Introduction to modelling

VARs and monetary policy exercise Our data is contained in the Excel spreadsheet US and Canada.xls. The spreadsheet comprises, among others, the following data series for Canada (denoted by _CN) and the US (denoted by _US):1

• base rates (BR_CN and BR_US); • the three-month Canadian Treasury Bill interest rate (TREASURYB_CN); • an average 3-5-year Canadian government bond yield (GOVY_CN); • nominal Canadian M1, M2 and M3 (M1_CN, M2_CN and M3_CN); • consumer price indices (CPI_CN and CPI_US); • GDP deflators (GDPDEF_CN and GDPDEF_US); • real GDPs (RGDP_CN and RGDP_US); • real Canadian gross fixed capital formation (RINV_CN); • nominal and effective exchange rates (NOM_EER_CN and NOM_EER_US as well as

RE_CN and RE_US); • the volume of Canadian exports and imports (X_CN and IM_CN); • the Canadian unemployment rate (UN_CN); and • a commodity price index (PCM)

We constructed other series, such as inflation (INF_CN and INF_US), the interest-rate differential between Canada and the US (BR_CN_US), the exchange rate depreciation (DPN_CN and DPN_US), the output gap using the Hodrick-Prescott filter (GDP_GAP_CN and GDP_GAP_US) and the annual growth rate of (real) GDP (DGDP4_CN and DGDP4_US). Variables that start with an ‘L’ denote the natural logarithm of the series. The data are quarterly and span the period from 1950 Q1 to 2005 Q2, although not all series are available for such a long period. Recall from the presentation that it is important to estimate parameters in structural VARs by concentrating on a single policy regime. Any regime shift requires a different parameterisations. This important caveat may explain some of the counterintuitive results we may encounter in the following exercise.2 The main justification for this practice is that monetary policy shocks are deemed to be robust to the different identification schemes generated by the different monetary policy regimes. 1 Preparations VAR modelling in JMulTi is meant as a step-by-step procedure, where each task is related to a special panel. Once a model has been estimated, the diagnostic tests as well as the stability and the structural analyses use the results from the estimation. If changes in the model specification are made, these results are deleted and the model has to be re-estimated. In other words, only results related to any one model at a time are kept in the system. Hence, there should be no confusion regarding the model set-up while going through the analysis. Open JMulTi and wait for the programme to load. Of the seven options in the window on the left, select the second-to-last, VAR Analysis. Go to File, Import Data, change the file type to .xls and open

1 But we will not be using all the data series in the following analysis. 2 For example, many SVAR studies of US monetary policy shocks leave out the disinflationary period from 1979 to 1984, which clearly constitutes a different monetary policy regime.

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the file US and Canada.xls. The only thing we have to tell JMulTi when loading data is the start of the sample period, so please enter ‘1950 Q1’ when prompted. 2 Closed-economy estimation of monetary policy shocks Monetary economics focuses on the behaviour of prices, monetary aggregates, nominal and real interest rates and output. Vector autoregressions (VARs) have served as a primary tool in much of the empirical analysis of the interrelationships between those variables and for uncovering the impact of monetary phenomena on the real economy. A summary of the empirical findings in this literature can be found in, inter alia, Leeper et al. (1996) and Christiano et al. (1998). On the basis of extensive empirical VAR analysis, Christiano et al. (1998) derived stylised facts about the effects of a contractionary monetary policy shock. They concluded that plausible models of the monetary transmission mechanism should be consistent with at least the following evidence on prices, output and interest rates. Following a contractionary monetary policy shock:

(i) the aggregate price level initially responds very little; (ii) interest rates initially rise; and (iii) aggregate output initially falls, with a J-shaped response, with a zero long-run effect of the

monetary policy shock. Recall that monetary policy shocks are defined as deviations from the monetary policy rule that are obtained by considering an exogenous shock which does not alter the response of the monetary policy-maker to macroeconomic conditions. VAR models of the effects of monetary policy shocks have therefore exclusively concentrated on simulating shocks, leaving the systematic component of monetary policy unaltered. 3 Defining a monetary policy VAR model It is interesting to see how the specification of the closed-economy benchmark model of monetary policy shocks has developed over time. In the exercises below, we will broadly follow the historical development of this model. Initially, such models were estimated (and identified) on a rather limited set of variables, i.e., prices, output and money, and identified using a diagonal form on the variance-covariance matrix D of the structural shocks and a lower triangular form on the contemporaneous impact matrix F. As we will see, there have been significant changes in the empirical analysis since then. First, the position of the monetary-policy instrument in the ordering of the variables included in the VAR has been changing over time. Second, the monetary policy instrument is no longer a monetary aggregate. Most analysis now generally acknowledges that a (short-term) interest rate is the relevant monetary-policy instrument. Moreover, the nature of the identification restrictions has also changed from recursive systems to non-recursive schemes that attempt identification on the basis of postulated economic relationships. Finally, most industrial-country benchmark models of monetary policy now contain six or seven variables rather than the initial three. A key consideration before any estimation is the form the variables must have when they enter the VAR: should they enter in levels, gaps or first differences? The answer is simply ‘it depends’. It depends on what the VAR is going to be used for. For forecasting purposes, we must avoid potential spurious regressions that may result in spurious forecasts; for the purpose of identifying structural shocks (such as monetary policy shocks) we have to be careful about the stability of the VAR (i.e., whether it can be inverted), the reliability of our impulse response functions and the residuals. Q1. Before we estimate our model, what could we do to ensure unbiased estimates?

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Answer: We use OLS to estimate the VARs so if we are interested in forecasting, say, we need to ensure that all variables are stationary or cointegrated to avoid the spurious regression problem associated with unit roots. As shown by Toda and Yamamoto (1995) and Dolado and Lütkepohl (1996), if all variables in the VAR are either I(0) or I(1) and if a null hypothesis is considered that does not restrict elements of each of the Ai’s (i = 1, … , p), the usual tests have their standard asymptotic normal disturbances. Moreover, if the VAR order p ≥ 2, the t-ratios have their usual asymptotic standard normal distributions and they remain suitable statistics for testing the null hypothesis that a single coefficient in one of the parameter matrices is zero (while leaving the other parameter matrices unrestricted). This alleviates the spurious regression problem on the use of standard asymptotic normal disturbances. Given the results in Sims et al. (1990), potential non-stationarity in the VAR under investigation should not affect the model selection process. Moreover, maximum likelihood estimation procedures may be applied to a VAR model fitted to the levels even if variables have unit roots; hence, possible cointegration rejections are ignored. This is frequently done in (S)VAR modelling to avoid imposing too many restrictions, and we follow this approach here. 3.1 Sims’ (1980b) model: putting money first We follow the academic literature that has tried to identify monetary policy shocks. In common with that literature we use variables in levels (and in logs), see, e.g., Christiano et al. (1998), Favero (2001) and Leeper et al. (1996). Q2. Estimate an unrestricted VAR. We start our estimation with LM1_CN, LRGDP_CN and LCPI_CN, including also a constant. Select the three variables in that order, then confirm your selection. Remember that selection order sets variable order. The chosen variables in the order in which they were chosen appear in red in a window at the bottom of the right-hand side. Pressing Max selects the longest sample period (which should be 1961 Q1 to 2005 Q1 in this case). Note that we will be measuring the impact of monetary policy by positive shocks to the monetary aggregate, M1, i.e., expansionary monetary policy. The typical identifying assumption in much of Sims’ earlier work (such as Sims (1980b)) is that the monetary policy variable is unaffected by contemporaneous innovations in the other variables, that is, it is put ‘first’ in the VAR. This approach corresponds to the assumption that the money supply is predetermined and that policy innovations are exogenous with respect to the non-policy innovations. This corresponds to a situation in which the policy variable does not respond contemporaneously to output shocks, perhaps because of information lags in formulating policy. Q3. What should be the lag length of the VAR? Adding more lags improves the fit of the estimated model but reduces the degrees of freedom and increases the danger of over-fitting. An objective way to decide between these competing objectives is to maximise some weighted measures of these two parameters. This is how the Akaike information criterion (AIC), the Schwarz Bayesian criterion (SC) and the Hannan-Quinn criterion (HQ) work. These three statistics are measures of the trade-off of fit against loss of degrees of freedom so that the best lag length should minimise all of them.3 An alternative to the information criterion is to systematically test for the significance of each lag using a likelihood-ratio test (discussed in Lütkepohl, 1991, section 4.3). This is the approach favoured by Sims (1980a). Since a VAR of lag length n nests the same VAR of lag length (n - 1), the log-likelihood 3 Some programs maximise the negative of these measures.

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difference multiplied by the number of observations less the number of regressors in the VAR should be distributed as a χ2-squared distribution with 2k degrees of freedom, i.e., χ2(2k). In other words,

LR = (T - m) logΩn-1 - logΩn ~ χ2(2k) where T is the number of observations, m is the number of regressors in each equation of the unrestricted VAR and logΩn is the log-likelihood of the VAR with n lags. For each lag length, if there is no improvement in the fit from the inclusion of the last lag, then the difference in log-likelihoods should not be significantly different from zero. For example, we could use a general-to-specific methodology (some authors follow a specific-to-general approach, instead):

(i) we start with a high lag length (say, 12 lags for quarterly data);4 (ii) for each lag, say n, note its log-likelihood and then calculate the log-likelihood for a VAR of

lag (n-1); (iii) take the difference of the log-likelihoods; and (iv) this difference should be distributed as a χ2-distribution.5

It is important to note that the residuals from the estimated VAR should be well behaved, that is, there should be no problems with autocorrelation and non-normality. Thus, whilst the AIC or the SC may be good starting points for determining the lag-length of the VAR, it is still important to check for autocorrelation. If we find that there is autocorrelation for the chosen lag length, one ought to increase the lag-length until the problem disappears. Similarly, if there are problems with non-normality, a useful trick is to add exogenous variables to the VAR (they may correct the problem), including the use of dummy variables and time trends. At the same time, we should note that the specification of the monetary policy VAR and its statistical adequacy is an issue that has not received much explicit attention in the literature. In most of the applied papers, the lag length is either decided uniformly on an ad hoc basis, several different lag lengths are estimated for ‘robustness’, or the lag length is simply set on the basis of information criteria. Virtually none of the papers cited in this exercise actually undertake any rigorous assessment of the statistical adequacy of the estimated models. Q4. Test for the appropriate number of lags using the AIC, SC, HQ and the final prediction error (FPE) criteria. Answer: To select the endogenous lags, it may be helpful to use the information criteria. Set Endogenous max lags to 12, tick the box for a constant only and press Compute Infocriteria. A search is performed over the lags of the endogenous variables up to the maximum order. The output should be: OPTIMAL ENDOGENOUS LAGS FROM INFORMATION CRITERIA endogenous variables: LM1_CN LRGDP_CN LCPI_CN deterministic variables: CONST sample range: [1964 Q1, 2005 Q1], T = 165 optimal number of lags (searched up to 12 lags of levels): Akaike Info Criterion: 5 Final Prediction Error: 5 Hannan-Quinn Criterion: 3

4 Obviously, this will depend on the frequency of your data, so that if you have quarterly data you could start with say 10-12 lags, if you have annual data with 2-3 lags and if you have monthly data with 18-24 lags. 5 This depends on the number of variables in the VAR. For our case we have three variables, thus 2*k = 2*3 = 6.

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Schwarz Criterion: 2 The correct lag length will depend on the criteria or measure we use. This is typical of these tests and researchers often use the criterion most convenient for their needs. The SC criterion is generally more conservative in terms of lag length than the AIC criterion, i.e., it selects a shorter lag length than the AIC criterion. Q5. Does the chosen VAR have appropriate properties? Are the residuals normal and not autocorrelated? Is the VAR stable? In order to carry out tests for these properties, the model will have to be specified and estimated first. A useful tip is to start with the VAR with the minimum number of lags according to the information criteria (in this case two lags) and test whether there are problems with autocorrelation and non-normality. Under Specification, Specify VAR Model, set endogenous lags equal to two and go to the Estimation panel and select Estimated Model. In JMulTi, plots of autocorrelations, partial autocorrelations and cross-correlations are found in Model Checking, Residual Analysis, Correlation. The following three charts show autocorrelations as well as partial autocorrelations for each of the error series u1, u2 and u3. On several occasions, both u2 and u3 markedly exceed the ±2/√T bounds. Some of these residual autocorrelations give rise to concern as the ‘large’ autocorrelations occur at short lags, especially for u3. We can be more sanguine about residual autocorrelations reaching outside the ±2/√T bounds if these violations occur at ‘large’ lags, as is the case for u2, say.

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Answer: In JMulTi, tests for residual autocorrelation, non-normality and conditional heteroskedasticity are available for diagnostic checking of estimated VAR models. Having estimated the VAR(2) model, residual tests can be found under Model Checking, Residual Analysis. Select the tests you want in the Diagnostic Tests window (in our case, the portmanteau test (with 16 lags), the LM type-test for

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autocorrelation (with five lags), tests for normality and the univariate ARCH-LM test (with 16 lags)) and press Execute. 6 The portmanteau test checks the null hypothesis H0: E(utu′t-i) = 0, i = 1, … , h (there is no remaining autocorrelation at lags 1 to h) against the alternative that at least one autocovariance and, hence, one autocorrelation is non-zero, up to order h. The test may only be applied if the VAR does not contain exogenous variables. The null hypothesis of no residual autocorrelation is rejected for large values of the test statistic. An adjusted version of the test statistic may have better small-sample properties: PORTMANTEAU TEST (H0:Rh=(r1,...,rh)=0) Reference: Lütkepohl (1993), Introduction to Multiple Time Series Analysis, 2ed, p. 150. tested order: 16 test statistic: 185.5839 p-value: 0.0004 adjusted test statistic: 194.8473 p-value: 0.0001 degrees of freedom: 126.0000 Both the original as well as the adjusted test statistic clearly allow us to reject the null hypothesis that all autocovariances and, hence, all autocorrelations, up to order h are zero, indicating the presence of autocorrelation. This is borne out by the Breusch-Godfrey LM test for hth order residual autocorrelation, which assumes the following underlying AR(h) model of the residuals:

ut = B1ut-1 + … + Bhut-h + vt and tests the null hypothesis that B1 = B2 = … = Bh = 0 against the alternative hypothesis that B1 ≠ 0 or … or Bh ≠ 0. The null hypothesis is rejected if LMh is large and exceeds the critical value from the χ2(h)-distribution. Again, this test may be biased in small samples and therefore an F-version of the statistic is given which may perform better (LMFh): LM-TYPE TEST FOR AUTOCORRELATION with 5 lags Reference: Doornik (1996), LM test and LMF test (with F-approximation) LM statistic: 83.9031 p-value: 0.0004 df: 45.0000 LMF statistic: 1.9648 p-value: 0.0003 df1: 45.0000 df2: 449.0000 Just as in the case of the portmanteau test for autocorrelation, both the original as well as the adjusted test statistic allow us to reject the null hypothesis that all estimated Bi’s up to order h are zero, indicating the presence of autocorrelation. Remaining residual autocorrelation indicates a model defect. It may be worth trying a VAR model with higher lag orders in that case. 6 The value of h, i.e. the number of autocorrelations included, is obviously important. The size of the test may be unreliable if h is too small (primarily because the χ2-approximation to the null distribution may be very poor), and it may have reduced power if h is large and, hence, many ‘non-informative’ autocorrelations are included. For quarterly data, a convenient benchmark is to set the lag length for the portmanteau test as well as the univariate ARCH-LM test equal to 16 and the lag order of the multivariate ARCH-LM (if required) as well as the LM tests for autocorrelation equal to 5.

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The idea underlying the (multivariate) non-normality tests is to transform the residual vector such that its components are both standardised and independent and then check the compatibility of the third and fourth moments, in other words, skewness and kurtosis, with those of a normal distribution (Doornik and Hansen (1994)). Lütkepohl (1991) proposed an alternative way of computing standardised residuals, so a test statistic based on his approach is also given.7 Both testing approaches indicate that there are no problems with non-normality: TESTS FOR NONNORMALITY Reference: Doornik & Hansen (1994) joint test statistic: 5.8910 p-value: 0.4355 degrees of freedom: 6.0000 skewness only: 2.9446 p-value: 0.4002 kurtosis only: 2.9464 p-value: 0.4000 Reference: Lütkepohl (1993), Introduction to Multiple Time Series Analysis, 2ed, p. 153 joint test statistic: 5.9006 p-value: 0.4344 degrees of freedom: 6.0000 skewness only: 2.8385 p-value: 0.4172 kurtosis only: 3.0621 p-value: 0.3822 JARQUE-BERA TEST variable teststat p-Value(Chi^2) skewness kurtosis u1 3.6107 0.1644 0.2065 3.5697 u2 1.2060 0.5472 0.1962 3.1068 u3 0.9640 0.6175 -0.0976 3.3067 Rejection of the null hypothesis of normality may indicate that there are some outlying observations or that the error process is not homoskedastic. An ARCH-LM test indicates problems with the series u2 and u3, as the tests statistics associated with these series are statistically significant, indicating ARCH behaviour in the residual series: ARCH-LM TEST with 16 lags variable teststat p-Value(Chi^2) F stat p-Value(F) u1 13.2732 0.6527 0.9051 0.5646 u2 49.5994 0.0000 4.5054 0.0000 u3 29.5363 0.0206 2.2672 0.0057 Finally, we can investigate the issue of stability.

7 Doornik and Hansen (1994) use the square root matrix of the residual covariance matrix, while Lütkepohl (1991) uses the Choleski decomposition. As is well-known, analysis based on the Choleski decomposition may depend on the ordering of the variables.

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Answer: Since we want to obtain a vector-moving average (VMA) from the VAR, we have to ensure that our VAR is stable (and thus invertible). It is easy to check for stability in JMulTi. The top portion of the Output (save/print) window under Estimation shows the modulus of the eigenvalues of the reverse characteristic polynomial: modulus of the eigenvalues of the reverse characteristic polynomial: |z| = (15.1704 4.8395 1.5459 1.0094 1.0656 0.9875) The last entry is below one in magnitude; thus, this VAR has a problem with stability. In other words, it cannot be inverted. In light of the foregoing problems with autocorrelation and instability, we may want to consider adding more lags as well as deterministic regressors to the estimated VAR model, which would address the autocorrelation and stability problems respectively. Adding more lags seems to get rid of the problems with autocorrelation. In particular, having a total of three lags eliminates the problems with autocorrelation on the basis of the LM-type test but not the portmanteau test:8 PORTMANTEAU TEST (H0:Rh=(r1,...,rh)=0) Reference: Lütkepohl (1993), Introduction to Multiple Time Series Analysis, 2ed, p. 150. tested order: 16 test statistic: 160.9276 p-value: 0.0044 adjusted test statistic: 169.9171 p-value: 0.0010 degrees of freedom: 117.0000 LM-TYPE TEST FOR AUTOCORRELATION with 5 lags Reference: Doornik (1996), LM test and LMF test (with F-approximation) LM statistic: 58.8265 p-value: 0.0809 df: 45.0000 LMF statistic: 1.2860 p-value: 0.1087 df1: 45.0000 df2: 437.0000 In addition, the residual series are normal using the Doornik and Hansen (1994) test and we cannot reject the null hypothesis of normality using Lütkepohl’s (1993) approach: TESTS FOR NONNORMALITY Reference: Doornik & Hansen (1994) joint test statistic: 6.3197 p-value: 0.3884 degrees of freedom: 6.0000

8 This is, unfortunately, a common problem. The response to this quandary is mixed: many researchers pick the test result that best suits their purposes; others have a favourite test, while others still may continue until they have a test result that is robust in both tests. Note that simulation results in Hall and McAleer (1989) indicate that the LM test often has higher power than the portmanteau test.

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skewness only: 2.4140 p-value: 0.4910 kurtosis only: 3.9057 p-value: 0.2718 Reference: Lütkepohl (1993), Introduction to Multiple Time Series Analysis, 2ed, p. 153 joint test statistic: 6.4775 p-value: 0.3719 degrees of freedom: 6.0000 skewness only: 2.4516 p-value: 0.4841 kurtosis only: 4.0259 p-value: 0.2587 But the estimated VAR(3) is still unstable, as can be seen from the second to last modulus of the eigenvalues of the reverse characteristic polynomial, which is less than one in magnitude: modulus of the eigenvalues of the reverse characteristic polynomial : |z| = (2.1527 2.1527 3.3418 1.8351 1.8351 1.2656 1.0746 0.9872 1.0091) Adding a time trend helps the stability of the VAR (verifying this is left as an exercise).9 Note that adding a time trend will not always correct instability. Note also that in this case, the inclusion of a trend fortuitously helps with the Doornik and Hansen (1994) serial correlation test. It is worth spending a bit of time investigating the individual coefficient estimates. In JMulTi, VAR output is generated in matrix and text form. The matrix form is a particularly intuitive way of presenting the estimation results. The matrix panel displays first the endogenous, then the exogenous (if present) and finally the deterministic regressors. This way of presenting the results reflects the mathematical notation to make clear what type of model was actually estimated. By right-clicking on the respective coefficient tables one can increase or decrease the precision of the numbers. By clicking on the respective buttons, it is possible to display the estimated coefficients, their standard deviations or their t-statistics. Even though we estimate a relatively low-dimensioned three-variable VAR with three lags and two deterministic regressors (a constant and trend) using 174 data points, the model may already be overparametrised. Using the empirical results on the viability of the usual asymptotic distribution of test statistics by Toda and Yamamoto (1995) and Dolado and Lütkepohl (1996), we note that just nine of the 33 estimated parameters are statistically significant at the 5% level:

9 Although Sims (1980) recommends against the use of a deterministic trend in VAR analysis, we decided not to follow this advice.

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3.2 Forecasting VARs in JMulti While not part of this exercise, we should briefly touch upon forecasting VARs in JMulTi, which is extremely straightforward as well. Say we are interested in forecasting any (or all) of the variables in the small three-variable VAR that we have just estimated. Go to the Forecasting panel and Forecast Variables. We assume that we are interested in (dynamic) forecasts over a three-year period.10 This would translate into a forecast horizon of 3*4 = 12 quarters. You will note that the screen changed after entering a forecast horizon of more than one period. This is because we have to tell JMulTi what the values for the deterministic as well as exogenous regressors are. In our case, we have two deterministic variables, which are the constant and a linear trend. Indeed, the forecasted values over the twelve-year period are an invariant value of one for the constant and a linearly increasing trend. Clicking on one of the three variables will select that variable to forecast, while clicking on variables while keeping the Ctrl-key pressed down allows for the selection of more than one variable.

10 Forecasts are based on conditional expectations assuming independent white-noise errors and are computed recursively.

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Pressing on Forecast generates the required forecast, resulting in text output on the Text (save/print) tab as well as the following graph:

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4 VAR identification We are now ready to attempt identification of the structural VAR and, by so doing, attempt to identify monetary policy shocks. In JMulTi, we have an underlying structural equation of the form: Ayt = C(L)yt + But (1) where the structural shocks ut are normally distributed, i.e., ut ∼ N(0, I). Unfortunately, we cannot estimate this equation directly due to identification issues, but instead we have estimated an unrestricted VAR of the form: yt = A-1C(L)yt + A-1But (2) Matrices A, B and the Cj’s are not separately observable. So how can we recover equation (1) from (2)? The solution is to impose restrictions on our VAR to identify an underlying structure, but what kind of restrictions are these? Economic theory can sometimes tell us something about the structure of the system we wish to estimate. As economists, we must interpret these structural or theoretical assumptions into restrictions on the VAR. Such restrictions can include, for example:

(i) a causal ordering of shock propagation, e.g., the Choleski decomposition; (ii) that nominal variables have no long run effect on real variables; and (iii) the long-run behaviour of variables, e.g., the real exchange rate is constant in the long-run.

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There are many types of restrictions that can be used to identify the structural VAR. JMulTi allows you to impose two of them. One type imposes restrictions on the short-run behaviour of the system, whereas the other type imposes restrictions on the long-run. JMulTi does not allow both types of restrictions at the same time. 5 Imposing short-run restrictions To impose short-run restrictions in JMulTi, we use equation (2): yt = A-1C(L)yt + A-1But We estimate the random stochastic residual A-1But from the residual εt of the estimated VAR: A-1But = εt (3) Reformulating equation (3), we have A-1Butut′B′(A-1)′ = εtεt′, and, since E(utut′) = I, we have: A-1BB′(A-1)′ = E(εtεt′) = Ω (4) Equation (4) says that for K variables in yt, the symmetry property of E(εtεt′) imposes K(K + 1)/2 restrictions on the 2K2 unknown elements in A and B. Thus, an additional K(3K - 1)/2 restrictions must be imposed on A and B to identify the full model. JMulTi requires that such restriction schemes must be of the form: Aεt = But (5) This is also known as the AB model. In particular, JMulTi offers three versions of the AB model:

(ii) an A model, where B = I (in which case Aεt = ut); (iii) a B model where A = I (in which case εt = But); (iv) and a general AB model where restrictions can be placed on both matrices.

By imposing structure on matrices A and B, we impose restrictions on the structural VAR in equation (1). An example of this specification using the Choleski decomposition identification scheme I from the presentation is:

A = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

101001

3231

21

aaa , B =

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

33

22

11

000000

bb

b

In our example, we have a VAR with three endogenous variables, requiring a total of 12 = 3(3*3 - 1)/2 restrictions. Counting restrictions in the A and B matrices above, we have nine zero restrictions (three in matrix A and six in matrix B) as well as another three unitary restrictions on the diagonal of matrix A, giving us the required total of 12 restrictions. Q6. Impose the Choleski decomposition, which assumes that shocks or innovations are propagated in the order of LM1_CN, LRGDP_CN and LCPI_CN. Answer: To impose the restriction above in matrix format, we select SVAR and SVAR Estimation from the VAR window menu. Then we need to choose between the SVAR AB model or the Blanchard and Quah long-run-type restriction:

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The identifying restriction is imposed in terms of the ε’s, which are the residuals from the VAR estimates, and the u’s, which are the fundamental or ‘primitive’ random (stochastic) errors in the structural system. To use JMulTi’s matrix-form restrictions, define the two matrices A and B with the following values:

A =⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

101001

*** , B =

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

**

*

000000

If the AB model is chosen, specify the kind of model you wish to work with (A, B or AB) and impose the restrictions by clicking on the relevant elements of the matrices activated in the panel. In the most general case, JMulTi allows you to change all elements in the A matrix as well as the diagonal elements in the B matrix. Ticking on Edit coefficients manually allows you to enter specific coefficient values. In order to impose the (lower-triangular) Choleski decomposition, select the Specify general case option and double-click on the elements below the diagonal in the A matrix until all of them come up with an asterisk. Executing this setup of the A and B matrices yields the following output: This is an AB-model Step 1: Obtaining starting values from decomposition of correlation matrix... Iterations needed for correlation matrix decomposition: 11.0000 Vector of rescaled starting values: -0.0224 0.0060 0.1224 0.0206 0.0074

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0.0049 Step 2: Structural VAR Estimation Results ML Estimation, Scoring Algorithm (see Amisano & Giannini (1992)) Convergence after 1 iterations Log Likelihood: 2191.8840 Structural VAR is just identified Estimated A matrix: 1.0000 0.0000 0.0000 -0.0224 1.0000 0.0000 0.0060 0.1224 1.0000 Estimated standard errors for A matrix: 0.0000 0.0000 0.0000 0.0273 0.0000 0.0000 0.0181 0.0503 0.0000 Estimated B matrix: 0.0206 0.0000 0.0000 0.0000 0.0074 0.0000 0.0000 0.0000 0.0049 Estimated standard errors for B matrix 0.0011 0.0000 0.0000 0.0000 0.0004 0.0000 0.0000 0.0000 0.0003 A^-1*B 0.0206 0.0000 0.0000 0.0005 0.0074 0.0000 -0.0002 -0.0009 0.0049 SigmaU~*100 0.0426 0.0010 -0.0004 0.0010 0.0055 -0.0007 -0.0004 -0.0007 0.0025 end of ML estimation Note that the ML estimates of the structural parameters also enable us to obtain an estimate of the contemporaneous impact matrix A-1B linking εt and ut (see equation (4)). Knowledge of this matrix allows us to recover the structural shocks from the estimated residuals. 6 Generating impulse responses and forecast error variance decompositions Two useful outputs from VARs are the impulse response function and the forecast error variance decomposition. Impulse responses show how the different variables in the system respond to (identified) shocks, i.e., they show the dynamic interactions between the endogenous variables in the VAR(p) process. Since we have ‘identified’ the structural VAR, the impulse responses will be depicting the responses to the structural shocks. In other words, once the structural model has been identified and estimated, the effects of the structural shocks ut can be investigated through an impulse response analysis. We do this because the results of the impulse response analysis are often more informative than the structural parameter estimates themselves.

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The same is true for forecast error variance decompositions, which are also popular tools for interpreting VAR models. While impulse response functions trace the effect of a shock to one endogenous variable onto the other variables in the VAR, forecast error variance decompositions (or variance decompositions in short) separate the variation in an endogenous variable into the contributions explained by the component shocks in the VAR. In other words, the variance decomposition tells us the proportion of the movements in a variable due to its ‘own’ shocks versus shocks to the other variables. Thus, the variance decomposition provides information about the relative importance of each random shock in affecting the variables in the VAR. In much empirical work, it is typical for a variable to explain almost all of its forecast error variance at short horizons and smaller proportions at longer horizons. Such a response is not unexpected, as the effects from the other variables are propagated through the reduced-form VAR with lags. Q7. Generate impulse response functions for the identification scheme in our example. Answer: Select SVAR and the newly highlighted option SVAR IRA (SVAR impulse response analysis). Now that we have identified the SVAR, this option has become available. A new screen will open up. We first have to Specify IRA, which concerns the number of periods (60) and the nature of the confidence intervals. A second window, Display Impulse Responses, allows us to fine-tune the exact nature of the charts. We would like to look at all impulse responses, so we select (by holding down the Ctrl-key while selecting series) all impulses and all responses. The following figure gives the responses of the three variables in the VAR to the identified structural shocks, together with 95% Hall bootstrap intervals based on 1,000 bootstrap replications:11

11 1,000 bootstrap replications may be a bit on the low side for reliable inference. But a higher number of bootstrap replications may be rather computationally demanding (least of all on available memory when VARs with several variables and long lags are considered) and therefore affect computing time. For reliable confidence intervals, the bootstrap literature recommends that the number of replications should be of the order or 2,000 or more.

18

In the light of Christiano et al.’s (1998) three stylised facts about the effects of contractionary monetary policy shocks, let us analyse the impact of an (expansionary) shock to the monetary aggregate M1 on the three variables in the VAR. These are given in the first column of the above chart. Money obviously increases as a result of a positive shock to itself. Output shows the required J-shaped response, albeit the other way, as the monetary policy shock is expansionary rather than contractionary. The long-run response of output to the expansionary monetary shock is zero. The aggregate price level responds after one year. Whether this corresponds to Christiano et al.’s (1998) first stylised fact of the price level initially responding very little is open to debate. Q8. Generate forecast error variance decompositions for the identification scheme in our example. Answer: Select SVAR and the newly highlighted option SVAR FEVD (SVAR forecast error variance decomposition). Now that we have identified the SVAR, this option has become available. A new screen will open up. We first have to Specify FEVD, which concerns the number of periods (20). The following table gives the variance decompositions of the three variables in the VAR to the identified structural shocks (at four-quarter intervals to conserve space): SVAR FORECAST ERROR VARIANCE DECOMPOSITION Proportions of forecast error in "LM1_CN" accounted for by: forecast horizon LM1_CN LRGDP_CN LCPI_CN 1 1.00 0.00 0.00 4 0.99 0.01 0.00 8 0.99 0.01 0.00 12 0.99 0.01 0.00 16 0.98 0.01 0.01 20 0.97 0.01 0.02 Proportions of forecast error in "LRGDP_CN" accounted for by: forecast horizon LM1_CN LRGDP_CN LCPI_CN 1 0.00 1.00 0.00 4 0.17 0.81 0.02 8 0.23 0.73 0.04 12 0.25 0.70 0.04 16 0.27 0.68 0.05 20 0.28 0.67 0.05 Proportions of forecast error in "LCPI_CN" accounted for by: forecast horizon LM1_CN LRGDP_CN LCPI_CN 1 0.00 0.03 0.97 4 0.02 0.02 0.97 8 0.11 0.02 0.87 12 0.26 0.08 0.66 16 0.39 0.15 0.46 20 0.48 0.21 0.31 To begin with, output and prices predict almost none of the variance of money. Of much more interest, however, is the effect of nominal on real variables, as it is sometimes found that monetary policy shocks explain only a small part of the variance of output. Overall, the M1 monetary aggregate accounts for a

19

little more than a quarter (28%) of the variation in output after five years (20 quarters), while shocks to real output themselves account for two-thirds of the output variance at that horizon. Two comments are warranted. As pointed out by Bernanke (1996), this does not mean that all monetary policy has been unimportant. First, it could be the case that anticipated monetary policy, or more generally, the systematic part of monetary policy, decreases the variance of output and inflation. Second, the variance decomposition does not tell us about the potential effects of monetary policy surprises (while the impulse response function does), only about the combination of the potential effect with the actual monetary policy shocks for that particular sample. 6.1 Putting money last (Sims (1992)) In later work by Sims and others, monetary policy is put last instead, which means that monetary policy is potentially affected by, but does not affect, contemporaneous macroeconomic variables. This corresponds to the assumption that policy shocks have no contemporaneous impact on output – and represents a Choleski decomposition with output and prices ordered before the monetary policy variable. How reasonable such an assumption might be clearly depends on the unit of observation: in annual data, the assumption of no contemporaneous effect would be implausible; with monthly data, it might be much more plausible. Theoretically, changing the order of the variables does make a difference. Recall from the presentation that the structural shocks, ut, and the observed shocks, εt, are related by: ut = Fεt (6) With money put first, we have the following system:

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

+++=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

CPIt

GDPt

Mt

GDPt

Mt

Mt

CPIt

GDPt

Mt

CPIt

GDPt

Mt

fffff

f

fffff

f

uuu

εεεεε

ε

εεε

33321

31

221

21

111

1

333231

2221

111

000

(7)

We can see that the first two shocks are unaffected by the third shock contemporaneously. This implies, for example, that a monetary policy shock takes time to affect the economy. On the other hand, if we put money last, we get the following system:

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

+++=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

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⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

1333231

2221

11

1333231

2221

11

1

000

Mt

CPIt

GDPt

CPIt

GDPt

GDPt

Mt

CPIt

GDPt

Mt

CPIt

GDPt

fffff

f

fffff

f

uuu

εεεεε

ε

εεε

(8)

The order of the variables therefore should matter, as the variable ordered last has no contemporaneous impact on the other variables in the system. Q9. Estimate an unrestricted VAR. We start our estimation with LRGDP_CN, LCPI_CN and LM1_CN, including also a constant. Select the three variables in that order, then confirm your selection. Remember that selection order sets variable order. The chosen variables in the order in which they were chosen appear in red in a window at the bottom of the right-hand side. Pressing Max selects the longest sample period. Note that we are still measuring the impact of monetary policy by positive shocks to the monetary aggregate, M1. Answer: We estimate the VAR with three lags, a constant plus a trend.

20

In terms of the results of the impulse response analysis, prices react within one year (and rather persistently) to monetary policy, output responds in the short run, in the long run (from ten years after the shock onwards) prices start adjusting and the significant effect on output vanishes after some 30 quarters. There is no strong evidence for the endogeneity of money. This is easily checked by looking at the estimated parameters in F (the A matrix in JMulTi’s output) and by analysing forecast error variance decompositions. The estimated coefficients in the last row of the A matrix are insignificant, so structural shocks to output and prices do not enter into the structural monetary policy shocks. FEVDs are very similar to the case where money is put first in the VAR, and show that macroeconomic variables play a very limited role in explaining the variance of the forecasting error of money, while money instead plays an important role in explaining fluctuations of both the other macroeconomic variables. 7 What monetary measure contains the most information? There is a long tradition of identifying monetary policy shocks with statistical innovations to monetary aggregates like base money (M0), M1 and M2. But which measure of money should we choose? Walsh (1998, p. 12) provides a first clue. He illustrates correlations between the detrended log of real US GDP and three different US monetary aggregates, each in detrended log form as well, and finds that the correlations with real output change substantially as one moves from M0 to M2. The narrow measure M0 is positively correlated with real GDP at both leads and lags. In contrast, M2 is positively correlated at lags but negatively correlated at leads. The larger correlations between GDP and M2 arise in part from the endogenous nature of an aggregate such as M2, depending as it does on the behaviour of the banking sector as well as that of the non-bank private sector (King and Plosser (1984), Coleman (1996)). In order to see what effects different monetary aggregates have, we can estimate the foregoing three-variable VAR with other monetary measures, in particular M2 and M3.

21

Q10. Estimate an unrestricted VAR. We start our estimation with LRGDP_CN, LCPI_CN and LM2_CN, including also a constant. Select the three variables in that order, then confirm your selection. Remember that selection order sets variable order. The chosen variables in the order in which they were chosen appear in red in a window at the bottom of the right-hand side. Pressing Max selects the longest sample period. Note that we are now measuring the impact of monetary policy by positive shocks to the monetary aggregate, M2. Answer: We estimate the VAR with four lags, a constant and a trend. The results of the impulse response analysis remain broadly unaffected: prices again react immediately (and persistently) to monetary policy and output responds in the (very) short run (up to one year), but shows no long-run effect. There is no strong evidence for the endogeneity of money. This is easily checked by looking at the estimated parameters in F (the A matrix in JMulTi’s output) and by analysing forecast error variance decompositions. The estimated coefficients in the last row of the A matrix for output and prices are insignificant, so structural shocks to output and prices do not enter into the structural monetary policy shocks: Estimated A matrix: 1.0000 0.0000 0.0000 0.0932 1.0000 0.0000 -0.1183 0.0646 1.0000 Estimated standard errors for A matrix: 0.0000 0.0000 0.0000 0.0554 0.0000 0.0000 0.0942 0.1398 0.0000 The next exercise asks you to use the series LM3_CN. Note that the list of available time series only contains M3_CN at the moment. In other words, we need to generate the logarithm of this series. Select the series M3_CN by clicking on it. Various tasks on the selected variable can be accessed by a menu that pops up when you right-click on the selected variable in the time series selector. We want to calculate the logarithm of M3_CN, so we select Transform, which opens up a new window. We click in the box next to logarithm, which will apply a logarithmic transformation to the selected variable in levels. Click OK and a new series appears in the time series selector, called M3_CN_log. As we want the name of the variable to be consistent with the other monetary aggregates, we need to rename the series. Right-click on M3_CN_log and select Rename. We can now give M3_CN_log its new name, which is LM3_CN. Q11. Estimate an unrestricted VAR. We start our estimation with LRGDP_CN, LCPI_CN and LM3_CN, including also a constant. Select the three variables in that order, then confirm your selection. Remember that selection order sets variable order. The chosen variables in the order in which they were chosen appear in red in a window at the bottom of the right-hand side. Pressing Max selects the longest sample period. Note that we are now measuring the impact of monetary policy by positive shocks to the monetary aggregate, M3. Answer: We estimate the VAR with three lags, a constant and a trend. This time around, the results of the impulse response analysis are somewhat different: prices again react immediately but only for about 20 quarters and – except for a brief dip after five years - output does not respond in either the short- or the long-run. There is again no strong evidence for the endogeneity of money. This is easily checked by looking at the estimated parameters in F (the A matrix in JMulTi’s output) and by analysing forecast error variance decompositions. The estimated coefficients in the last row of the A matrix are insignificant, so structural shocks to output and prices do not enter into the structural monetary policy shocks.

22

But it is most informative to look at the variance decompositions. Recall that M1 accounted for slightly more than a quarter of the forecast variance of real output. What are the equivalent percentages for M2 and M3? SVAR FORECAST ERROR VARIANCE DECOMPOSITION Proportions of forecast error in "LRGDP_CN" accounted for by: forecast horizon LRGDP_CN LCPI_CN LM2_CN 1 1.00 0.00 0.00 4 0.96 0.00 0.04 8 0.96 0.00 0.04 12 0.96 0.01 0.03 16 0.95 0.02 0.03 20 0.94 0.03 0.03 Proportions of forecast error in "LM2_CN" accounted for by: forecast horizon LRGDP_CN LCPI_CN LM2_CN 1 0.01 0.00 0.99 4 0.07 0.04 0.88 8 0.36 0.06 0.58 12 0.57 0.04 0.38 16 0.68 0.03 0.29 20 0.75 0.02 0.24 SVAR FORECAST ERROR VARIANCE DECOMPOSITION Proportions of forecast error in "LRGDP_CN" accounted for by:

23

forecast horizon LRGDP_CN LCPI_CN LM3_CN 1 1.00 0.00 0.00 4 0.99 0.01 0.00 8 0.98 0.02 0.00 12 0.97 0.02 0.01 16 0.97 0.02 0.01 20 0.97 0.02 0.02 Proportions of forecast error in "LM3_CN" accounted for by: forecast horizon LRGDP_CN LCPI_CN LM3_CN 1 0.00 0.01 0.98 4 0.10 0.02 0.89 8 0.28 0.04 0.68 12 0.43 0.05 0.52 16 0.52 0.06 0.42 20 0.59 0.06 0.35 We can see that the percentage of variation explained by M2 and M3 drops drastically to three and two percent respectively. For some of these reasons, broad monetary aggregates have been substituted by narrower aggregates, such as bank reserves, for which it is easier to identify shocks mainly driven by the behaviour of the monetary policy maker. 8 What is the monetary policy instrument? How one measures monetary policy is a critical issue in the empirical literature and is a topic of open and ongoing debate. What is the monetary policy instrument: a short interest rate or a (narrow) monetary aggregate? In general, there is no clear choice for the monetary policy variable. If monetary policy were framed in terms of strict targets for the money supply, for a specific measure of banking sector reserves or for a particular short-term interest rate, then the definition of the monetary policy variable might be straightforward. In general, however, several candidate measures of monetary policy will be available, all depending in various degrees on both policy actions and non-monetary policy disturbances. What constitutes an appropriate candidate for the monetary policy instrument, and how this instrument depends on non-policy disturbances, will depend on the operating procedures the monetary authority is following as it implements monetary policy. McCallum (1983) argues that the correct measure of the monetary policy shock is the residual in a reaction function for the monetary-policy instrument:

monetary policy instrument = f(lagged macroeconomic data) + policy shock The crucial assumption is that the policy instrument does not depend on contemporaneous macroeconomic data. The money stock most likely does, since money demand is probably affected by shocks to income and prices (as well as policy shocks to the interest rate). This makes the innovation in the money stock a mixture of the monetary policy and other shocks. One way of selecting between a short-term interest rate and a narrow monetary aggregate would be to follow the analysis in Bernanke and Blinder (1992). They addressed this problem from three different angles, of which the first two can be easily replicated. In particular, they concluded that in the US, the federal funds rate dominated both money and bill and bond rates in forecasting real variables. The argument goes as follows. If a (short-term) interest rate is a measure of monetary policy and if monetary policy affects the real economy, the interest rate should be a good reduced-form predictor of major macroeconomic variables. Studying the information content of the short-term interest rate, they found that the federal funds rate is markedly superior to both monetary aggregates and most other

24

interest rates as a forecaster of the economy. In their second approach, they argued that if a short-term interest rate measures monetary policy, then it should respond to the central bank’s perception of the state of the economy. The second test, therefore, is to estimate monetary-policy reaction functions explaining movements in the short-term interest rate by lagged target variables. They found that their estimated reaction functions using the short-term interest rate gave much more reasonable responses to inflation and unemployment shocks. Other theoretical approaches to how the monetary policy instrument is chosen include Sims (1992), who argued that expansionary shocks to monetary policy should drive output up and lead to opposite movements in the money stock and the interest rate. This makes Sims choose the interest rate rather than M1 as the policy instrument since positive shocks to M1 in his analysis led to an increase in the interest rate and a decline in output. We will therefore now include a short-term interest rate into our analysis. Q12. Estimate an unrestricted VAR. We start our estimation with LCPI_CN, LRGDP_CN, BR_CN and LM1_CN as well as a constant. Select the four variables in that order, then confirm your selection. Remember that selection order sets variable order. The chosen variables in the order in which they were chosen appear in red in a window at the bottom of the right-hand side. Pressing Max selects the longest sample period. Note that we can now measure the impact of monetary policy by either changes to the base rate, BR_CN, or the monetary aggregate, LM1_CN.12 The idea is to examine the robustness of the previous results after identifying the part of money that is endogenous to the interest rate. Moreover, it allows one to compare the impact of monetary and interest rate shocks to observe how both instruments affect the variables of interest in our system. Answer: We estimate the VAR with three lags, a constant and a trend. The following chart gives the associated impulse response functions:

12 But keep in mind that a unit shock to the base rate is an increase in interest rates and therefore a contractionary monetary policy shock, while a unit shock in the money supply corresponds to an increase in the money supply and thus an expansionary monetary policy shock.

25

Just as observed by Sims (1992), we also find that monetary aggregate shocks do not look like monetary policy shocks in their effects, whereas interest rate shocks do resemble monetary policy shocks in their effects. In his original analysis, Sims (1992) also showed that these shocks had smooth, slow effects on prices and that they produced predictable movements in the money stock.13 In addition, Sims (1980b) showed that the fraction of the forecasting error variance in US output that can be attributed to money stock innovations is much lower when an interest rate is added to a VAR of money, prices and output. The idea is to see the robustness of the above results after identifying that part of money which is endogenous to the interest rate. This finding is equally borne out for Canadian data. The proportion of forecast error in output accounted for by the money supply drops from 28% without a short-term interest rate to single-digit figures after the inclusion of the base rate. After 20 quarters, the short-term interest rate accounts for slightly more than half of the forecast error in output: Proportions of forecast error in "LRGDP_CN" accounted for by: forecast horizon LCPI_CN LRGDP_CN BR_CN LM1_CN 1 0.03 0.97 0.00 0.00 4 0.04 0.84 0.04 0.08 8 0.04 0.61 0.26 0.09 12 0.05 0.47 0.41 0.07 16 0.05 0.39 0.49 0.06 20 0.06 0.35 0.54 0.05 Following Sims’ (1992) arguments, we might also be tempted to conclude that the short-term interest rate – rather than the narrow monetary aggregate – is the monetary-policy instrument in Canada. 9 Some puzzles in recursively identified structural VARs 9.1 The output, liquidity and price puzzles Eichenbaum (1992) presents a comparison of the estimated effects of monetary policy in the US using alternative measures of policy shocks, discussing how different choices can produce puzzling results (at least puzzling relative to certain theoretical expectations). He based his discussion on the results obtained from a VAR containing two different combinations of the same four variables as in Sims (1992): the price level, output, M1 as a measure of the money supply and the federal funds rate as a measure of short-term interest rates. He considered interpreting shocks to M1 as policy shocks versus the alternative of interpreting federal funds-rate shocks as policy shocks. He found that a positive innovation to M1 is followed by an increase in the federal funds rate and a decline in output (the latter being coined the ‘output puzzle’). This result is puzzling if M1 shocks are interpreted as measuring the impact of monetary policy. An expansionary monetary policy would be expected to lead to increases in both M1 and output. The interest rate was found to rise after a positive M1 shock, also a potentially puzzling result; a standard model in which money demand varies inversely with the nominal interest rate would suggest that an increase in money supply would require a decline in the nominal interest rate to restore money-market equilibrium. Q13. Estimate an unrestricted VAR. We start our estimation with LCPI_CN, LRGDP_CN, LM1_CN and BR_CN as well as a constant.14 Select the four variables in that order, then confirm your

13 As indirect evidence, Sims (2007) also cites the fact that the money supply is close to a random walk, which is what you would expect if the monetary authority is systematically smoothing interest rates and money demand is shifting around in response to various private-sector disturbances. 14 The empirical results are robust to the other ordering of the variables, namely LM1_CN, LCPI_CN, LRGDP_CN and BR_CN.

26

selection. Remember that selection order sets variable order. The chosen variables in the order in which they were chosen appear in red in a window at the bottom of the right-hand side. Pressing Max selects the longest sample period. Note that we can now measure the impact of monetary policy by either innovations to the base rate, BR_CN, or the monetary aggregate, LM1_CN. Answer: We estimate the VAR with three lags, a constant and a trend. The results from the impulse responses and forecast error variance decompositions raise a number of issues. The following chart gives the associated impulse response functions:

Starting with the responses to the money supply shock (which is given by the third column), we note the absence of an output puzzle for Canadian data: expansionary monetary policy is followed by an increase in output. Note, however, that the response to money innovations gives rise to the ‘liquidity puzzle’: the interest rate declines very slightly contemporaneously in response to a monetary policy shock to start increasing afterwards. When Eichenbaum (1992) used innovations in the short-term interest rate as a measure of monetary policy actions (the fourth column), a positive shock to the federal funds rate represented a contractionary monetary policy shock. No output puzzle was found in this case: a positive interest-rate shock was followed by a decline in the output measure. Instead, the ‘price puzzle’ emerges: a contractionary policy shock is followed by a rise in the price level. The price puzzle is that in a VAR of output, prices, money, interest rates and perhaps some more variables, contractionary shocks to monetary policy lead to persistent price increases. This seems to hold not just in the US but also in several other countries, and is more pronounced if the policy instrument is taken to be a short interest rate than a monetary aggregate. It is often not statistically significant, but it is so common that it signals that the VAR might be misspecified. In fact, these effects are borne out for Canadian data. There is no output puzzle and we can see that the price puzzle is statistically significant for six quarters, i.e., a contractionary Canadian monetary policy shock is followed by a significant rise in the aggregate price level over six quarters.

27

Sims (1992) discusses how the price puzzle could be due to a missing element in the reaction function of the central bank. The most commonly accepted explanation for the price puzzle is that it reflects the fact that the variables included in the VAR do not span the information set available to the monetary policy-maker in setting the federal funds rate. Commodity prices (or other asset prices) may signal inflation expectations, so the central bank may react now by raising interest rates, which makes inflation somewhat lower than it would otherwise have been (but still positive). Since these prices tend to be sensitive to changing forecasts of future inflation, they serve to proxy for some of the Fed’s additional information. If commodity prices are excluded from the VAR, this may appear as monetary policy shocks having a positive effect on inflation. Turning to the variance decompositions, although little of the variation in money is predictable from past output and prices, a considerable amount (almost a third) becomes predictable when past short-term interest rates are included in the information set: SVAR FORECAST ERROR VARIANCE DECOMPOSITION Proportions of forecast error in "LM1_CN" accounted for by: forecast horizon LCPI_CN LRGDP_CN LM1_CN BR_CN 1 0.00 0.01 0.99 0.00 8 0.01 0.01 0.77 0.21 12 0.01 0.01 0.72 0.26 16 0.01 0.01 0.68 0.29 20 0.01 0.02 0.65 0.32 9.2 What about non-recursive identification? Recall that the main requirement of identification is to ensure that we can uniquely recover all the parameters in the Bj matrices and variance-covariance matrix D from the estimated Aj matrices and the variance-covariance matrix Ω of the estimated residuals. We do this by imposing the necessary n(n – 1)/2 additional restrictions. Recall from the presentation that it is only the overall number of identifying restrictions that matter; there is therefore nothing that necessarily requires identification restrictions to follow the Wold causal chain. But there is nothing from preventing us from imposing different identifying structure than the recursive one. Assume that data on the price level and real output emerge only with a lag and the monetary policymaker does not respond to changes in CPI and real GDP within the period. In addition, CPUI and real output do not respond within the period to changes in the short-term interest rate and the money supply. In the case of the above four-variable VAR (LRGDP_CN, LCPI_CN, BR_CN, LM1_CN), we could postulate the following system:

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

++++

++

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

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⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

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⎜⎜⎜⎜⎜

⎝

⎛

14443

134333231

2221

1211

14443

34333231

2221

1211

1 00

0000

Mt

BRt

Mt

BRt

CPIt

GDPt

CPIt

GDPt

CPIt

GDPt

Mt

BRt

CPIt

GDPt

Mt

BRt

CPIt

GDPt

ffffff

ffff

ffffff

ffff

uuuu

εεεεεε

εεεε

εεεε

(9)

Q14. Estimate a VAR with this identification system. Answer. Algebraically, the matrix contains the required n(n – 1)/2 = (4)(3)/2 = 6 zero restrictions required for identification. But JMulTi is unable to solve this system of equations as the first two equations have the same restrictions and are therefore indistinguishable. They can be premultiplied by any orthonormal (2 × 2) matrix and yet satisfy the same restrictions.

28

Q15. Since these two equations do not have separate interpretations, we normalise them arbitrarily by changing f12 in the first row from unrestricted to zero. Estimate a VAR with the following identification system:

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

1000010000100001

00

00000

BA ;

******

***

Such a restriction yields the following system:

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

++++

+=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

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⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

BRt

Mt

BRt

Mt

CPIt

GDPt

CPIt

GDPt

GDPt

Mt

BRt

CPIt

GDPt

Mt

BRt

CPIt

GDPt

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fff

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1 00

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(10)

The first line in (10) shows that the structural shocks to output are equivalent to the estimated output shocks or that structural output shocks are autonomous of the other variables in the system. The third equation in (10) contains contemporaneously all four of the traditional arguments of liquidity preference in an IS-LM model. For that reason, it could be labelled a ‘money demand’ equation. Similarly, the final equation in (10) could be regarded as a ‘money supply’ equation. The one over-identifying restriction cannot be rejected. While there are some advantages to abandoning the recursive assumption, there is also a substantial cost: a broader set of economic relations must be identified/ 9.3 Taking account of the price puzzle We now consider introducing alternative variables to our VAR. Taking inspiration from Sims (1992), we include the nominal exchange rate, LNOM_EER_CN, as an additional variable. As a forward-looking asset price, the exchange rate can also be expected to contain inflationary expectations. Following Sims’ (1992) argument, this should enlarge the information set of the monetary-policy maker and alleviate the price puzzle. Alternatively, you may want to perform the estimation using the commodity price index (PCM). We therefore now estimate our VAR with the following ordering: LCPI_CN, LRGDP_CN, BR_CN, LM1_CN and LNOM_EER_CN, that is, with the interest rate placed before the monetary aggregate. Q16. Determine the appropriate number of lags and whether to include a time trend in the VAR. Answer: Three lags plus a trend appear to be sufficient to guarantee a stable VAR without problems of serial correlation. Note, however, that the VAR does not have normal residuals! This is due to the introduction of the interest-rate term, which has experienced large movements. Indeed, looking at the residual diagnostics more closely, we can see the influence of large outliers in both the individual Jarque-Bera tests as well as the ARCH-LM tests for u5 (as well as u2 and u3), the residual associated with the base rate. This is, unfortunately, a serious problem.15 A Choleski decomposition yields the following impulse responses:

15 One way of dealing with this problem would be to introduce dummies to account for some of the larger outliers.

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The columns of interest are the third and fourth respectively. Examining first the columns, note that an interest rate shock leads to a fall in prices (although this is not clear on impact!), as well as a fall in output, and money. The impact on the exchange rate appears to be insignificant. The money shock increases prices and output (this is as before, although the magnitude as well as the time profile of the impulse responses have changed), it increases the interest rate (this suggests a Fisher-type effect rather than a standard Keynesian liquidity effect) and causes an insignificant depreciation of the currency. The Fisher effect has been documented in the US data by a number of authors. Note that a significant ‘price puzzle’ for Canada (the observation that on impact, an interest rate shock increases prices) is still present, but now only persists for two quarters. (Using LPCM instead of LNOM_EER_CN makes the price puzzle insignificant – you can do this as an additional exercise if you want to!) 10 Recursive versus non-recursive identification Sims (1986), for example, estimates a VAR including real GDP (y), real business fixed investment (inv), the GDP deflator (p), the money supply as measured by M1 (m), the unemployment rate (un) and the Treasury bill rate (r). An unrestricted VAR was estimated with four lags of each variable and a constant term. Sims obtained the 36 impulse response functions using a Choleski decomposition with the ordering y → inv → p → m → un → r. Some of the impulse response functions had reasonable interpretations. But the response of real variables to a money supply shock seemed unreasonable. The impulse responses suggested that a money supply shock had little effect on prices, output or the interest rate. Given a standard money demand function, it is hard to explain why the public would be willing to hold the extended money supply. Q17. Are these results borne out using a similar VAR for Canada? We estimate a VAR with the variables ordered as LRGDP_CN, LRINV_CN, LCPI_CN, LM1_CN, UN_CN and BR_CN. Three lags, a constant plus a trend appear to be sufficient to guarantee a stable VAR without problems of serial correlation using the LMF test. Note, however, that the VAR does not have normal residuals! A Choleski decomposition yields the following impulse responses (with 500 bootstrap replications):

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In contrast to Sims’ finding for the US, the responses of real variables to a money supply shock do not seem unreasonable with Canadian data. Sims proposes an alternative to the Choleski decomposition that is consistent with money-market equilibrium. He restricts the F matrix from the presentation (or the A matrix in JMulTi) such that:

⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

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64

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Note that there are 17 restrictions on the matrix, which is therefore overidentified; with six variables, exact identification requires only K(K – 1)/2 = 15 restrictions. Note also that the restrictions are no longer recursive. Imposing these restrictions, Sims identifies the following six relationships among the contemporaneous innovations: uyt = εyt + a12εinvt + a16εrt (11) uinvt = εinvt (12) upt = εpt + a31εyt + a32εinvt + a36εrt (13) umt = εmt + a41εyt + a43εpt + a46εrt (14) uunt = εunt + a51εyt + a52εinvt + a53εpt + a56εrt (15) urt = εrt + a64εmt (16) Sims views (14) and (16) as money demand and money supply functions respectively. In (16), the money supply should rise as the interest rate increases. The demand for money in (14) should be

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positively related to income and the price level and negatively related to the interest rate. Investment innovations in (12) are completely autonomous. Otherwise, Sims sees no reason to restrict the other equations in any particular fashion. Structural VAR Estimation Results ML Estimation, Scoring Algorithm (see Amisano & Giannini (1992)) Convergence after 1 iterations Log Likelihood: 2569.4532 Structural VAR is over-identified with 2.0000 degrees of freedom LR Test: Chi^2(2.0000): 18.0297 , Prob: 0.0001 Estimated A matrix: 1.0000 -0.0999 0.0000 0.0000 0.0000 -0.0049 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 -0.4485 0.1025 1.0000 0.0000 0.0000 -0.0056 -14.0244 0.0000 -32.7412 1.0000 0.0000 -0.0829 7.0545 2.4715 -1.7159 0.0000 1.0000 0.0369 0.0000 0.0000 0.0000 169.8628 0.0000 1.0000 Estimated standard errors for A matrix: 0.0000 0.1381 0.0000 0.0000 0.0000 0.1132 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 17.5999 1.8965 0.0000 0.0000 0.0000 0.0310 195.6438 0.0000 394.7980 0.0000 0.0000 1.4554 3.1851 1.0740 4.1272 0.0000 0.0000 0.0211 0.0000 0.0000 0.0000 683.9101 0.0000 0.0000 The estimated coefficients and their associated standard errors show how poorly estimated the coefficients in the A matrix are; in fact, only two of the 13 parameters are actually statistically significant at the 5% level (a51 and a52), so only the unemployment rate equation contains significant parameters. Moreover, the two overidentifying restrictions can actually be rejected. As could have been conjectured from the impulse response functions, Sims’ model does not seem to hold for Canada. 11 Extending monetary policy SVAR analysis to the open-economy Identifying exogenous monetary policy shocks in an open economy can lead to substantial complications relative to the closed-economy case, and papers that have examined the effects of monetary policy shocks in open economies have been distinctly less successful in providing accepted empirical evidence than the VAR approach in closed economies Q18. Canada is a small open economy, affected by the US. To reflect this we now add a US block. To add the US components, we can add the US variables of interest to our VAR as either endogenous or exogenous. Answer: For the endogenous VAR, estimate a VAR with variables in the following order: LCPI_US, LRGDP_US, BR_US, LCPI_CN, LRGDP_CN, BR_CN and LNOM_EER_CN, in other words, we order the US variables first (think why?). OPTIMAL ENDOGENOUS LAGS FROM INFORMATION CRITERIA endogenous variables: LCPI_US LRGDP_US BR_US LCPI_CN LRGDP_CN BR_CN LNOM_EER_CN deterministic variables: CONST sample range: [1964 Q1, 2005 Q1], T = 165

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optimal number of lags (searched up to 12 lags of levels): Akaike Info Criterion: 3 Final Prediction Error: 3 Hannan-Quinn Criterion: 2 Schwarz Criterion: 1 Information criteria suggest lag lengths between one and three, but you will find that JMulTi is no longer able to calculate residual test statistics with more than five lags. We therefore use the Hannan-Quinn criterion and estimate a VAR with two lags and a constant. The VAR is stable but the residual properties are autocorrelated and not normal. We will also have to limit the bootstrap replications to 500.

(We have not plotted the US variables to save space.) A number of observations are worth mentioning and there are some interesting results. In cases of non-normality, we have to be careful about the interpretation of the results, as asymptotic confidence intervals are likely to be suspect. But we have bootstrapped our confidence intervals, so we can be more confident about interpreting the results. US variables do appear to affect some Canadian variables: higher prices in the US lead to higher prices in Canada, and while higher output in the US does not increase Canadian prices, higher output in the US increases output in Canada. It is interesting to note that higher US GDP and higher US interest rates increase interest rates in Canada, suggesting that the Canadian monetary authorities may be looking at some US variables. It is also interesting to note that the exchange rate seems to react longer term to US interest rates as well as to its own shock.

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12 References Bernanke, B S (1996), ‘Comment on ‘What does monetary policy do?’’, Brookings Papers on

Economic Activity, Vol. 1996, No. 2, pages 69-73. Bernanke, B S and Blinder, A S (1992), ‘The federal funds rate and the channels of monetary

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Christiano, L J and Eichenbaum, M (1995), ‘Liquidity effects, monetary policy and the business cycle’, Journal of Money, Credit and Banking, Vol. 27, No. 4, pages 1113-36.

Christiano, L J, Eichenbaum, M and Evans, C L (1998), ‘Monetary policy shocks: what have we learned and to what end?’, in Taylor, J B and Woodford, M (eds.), Handbook of macroeconomics, Vol. 1A, Amsterdam, Elsevier Science, North-Holland, pages 65-148.

Coleman, W J (1996), ‘Money and output: a test of reverse causation’, American Economic Review, Vol. 86, No. 1, pages 90-111.

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Favero, C A (2001), Applied macroeconometrics, Oxford: Oxford University Press. Favero, C A (2009), ‘The econometrics of monetary policy: an overview’, in Mills, T C and Patterson,

K (eds.), Palgrave handbook of econometrics – Volume 2: Applied econometrics (forthcoming). Hall, A D and McAleer, M (1989), ‘A Monte Carlo study of some tests of model adequacy in time

series analysis’, Journal of Business & Economic Statistics, Vol. 7, No. 1, pages 95-106. Kim, S and Roubini, N (2000), ‘Exchange rate anomalies in the industrial countries: a solution with a

structural VAR approach’, Journal of Monetary Economics, Vol. 45, No. 3, pages 561-86. King, R G and Plosser, C I (1984), ‘Money, credit and prices in a real business cycle’, American

Economic Review, Vol. 74, No. 3, pages 363-80. Leeper, E M, Sims, C A and Zha, T (1996), ‘What does monetary policy do?’, in Brainard, W C and

Perry, G L (eds.), Brookings Papers on Economic Activity, Vol. 1996, No. 2, pages 1-63. Lucas, R E (1976), ‘Econometric policy evaluation: a critique’, Carnegie-Rochester Conference Series

on Public Policy, Vol. 1, pages 19-46. Lütkepohl, H (1991), Introduction to multiple time series analysis, Berlin: Springer-Verlag. McCallum, B T (1983), ‘A reconsideration of Sims’ evidence regarding monetarism’, Economic

Letters, Vol. 13, Nos. 2-3, pages 167-71. Rudebusch, G D (1998), ‘Do measures of monetary policy in a VAR make sense?’, International

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Sims, C A (2007), ‘Interview with Christopher Sims’, The Region – Banking and Policy Issues Magazine, The Federal Reserve Bank of Minneapolis, available from http://www.minneapolisfed.org/publications_papers/pub_display.cfm?id=3168.

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