Chapter4 Solutions

Solutions to the Review Questions at the End of Chapter 4

1. In the same way as we make assumptions about the true value of beta and not the estimated values, we make assumptions about the true unobservable disturbance terms rather than their estimated counterparts, the residuals.

We know the exact value of the residuals, since they are defined by . So we do not need to make any assumptions about the

residuals since we already know their value. We make assumptions about the unobservable error terms since it is always the true value of the population disturbances that we are really interested in, although we never actually know what these are.

2. We would like to see no pattern in the residual plot! If there is a pattern in the residual plot, this is an indication that there is still some “action” or variability left in yt that has not been explained by our model. This indicates that potentially it may be possible to form a better model, perhaps using additional or completely different explanatory variables, or by using lags of either the dependent or of one or more of the explanatory variables. Recall that the two plots shown on pages 157 and 159, where the residuals followed a cyclical pattern, and when they followed an alternating pattern are used as indications that the residuals are positively and negatively autocorrelated respectively.

Another problem if there is a “pattern” in the residuals is that, if it does indicate the presence of autocorrelation, then this may suggest that our standard error estimates for the coefficients could be wrong and hence any inferences we make about the coefficients could be misleading.

3. The t-ratios for the coefficients in this model are given in the third row after the standard errors. They are calculated by dividing the individual coefficients by their standard errors.

= 0.638 + 0.402 x2t - 0.891 x3t (0.436) (0.291) (0.763)

t-ratios 1.46 1.38 -1.17

The problem appears to be that the regression parameters are all individually insignificant (i.e. not significantly different from zero), although the value of R2 and its adjusted version are both very high, so that the regression taken as a whole seems to indicate a good fit. This looks like a classic example of what we term near multicollinearity. This is where the individual regressors are very

1/8 “Introductory Econometrics for Finance” © Chris Brooks 2008

closely related, so that it becomes difficult to disentangle the effect of each individual variable upon the dependent variable.

The solution to near multicollinearity that is usually suggested is that since the problem is really one of insufficient information in the sample to determine each of the coefficients, then one should go out and get more data. In other words, we should switch to a higher frequency of data for analysis (e.g. weekly instead of monthly, monthly instead of quarterly etc.). An alternative is also to get more data by using a longer sample period (i.e. one going further back in time), or to combine the two independent variables in a ratio (e.g. x2t / x3t ).

Other, more ad hoc methods for dealing with the possible existence of near multicollinearity were discussed in Chapter 4:

- Ignore it: if the model is otherwise adequate, i.e. statistically and in terms of each coefficient being of a plausible magnitude and having an appropriate sign. Sometimes, the existence of multicollinearity does not reduce the t-ratios on variables that would have been significant without the multicollinearity sufficiently to make them insignificant. It is worth stating that the presence of near multicollinearity does not affect the BLUE properties of the OLS estimator – i.e. it will still be consistent, unbiased and efficient since the presence of near multicollinearity does not violate any of the CLRM assumptions 1-4. However, in the presence of near multicollinearity, it will be hard to obtain small standard errors. This will not matter if the aim of the model-building exercise is to produce forecasts from the estimated model, since the forecasts will be unaffected by the presence of near multicollinearity so long as this relationship between the explanatory variables continues to hold over the forecasted sample.

- Drop one of the collinear variables - so that the problem disappears. However, this may be unacceptable to the researcher if there were strong a priori theoretical reasons for including both variables in the model. Also, if the removed variable was relevant in the data generating process for y, an omitted variable bias would result.

- Transform the highly correlated variables into a ratio and include only the ratio and not the individual variables in the regression. Again, this may be unacceptable if financial theory suggests that changes in the dependent variable should occur following changes in the individual explanatory variables, and not a ratio of them.


4. (a) The assumption of homoscedasticity is that the variance of the errors is constant and finite over time. Technically, we write

.

(b) The coefficient estimates would still be the “correct” ones (assuming that the other assumptions required to demonstrate OLS optimality are satisfied), but the problem would be that the standard errors could be wrong. Hence if we were trying to test hypotheses about the true parameter values, we could end up drawing the wrong conclusions. In fact, for all of the variables except the constant, the standard errors would typically be too small, so that we would end up rejecting the null hypothesis too many times. (c) There are a number of ways to proceed in practice, including- Using heteroscedasticity robust standard errors which correct for the problem by enlarging the standard errors relative to what they would have been for the situation where the error variance is positively related to one of the explanatory variables.

- Transforming the data into logs, which has the effect of reducing the effect of large errors relative to small ones.

5. (a) This is where there is a relationship between the ith and jth residuals. Recall that one of the assumptions of the CLRM was that such a relationship did not exist. We want our residuals to be random, and if there is evidence of autocorrelation in the residuals, then it implies that we could predict the sign of the next residual and get the right answer more than half the time on average!

(b) The Durbin Watson test is a test for first order autocorrelation. The test is calculated as follows. You would run whatever regression you were interested in, and obtain the residuals. Then calculate the statistic

You would then need to look up the two critical values from the Durbin Watson tables, and these would depend on how many variables and how many observations and how many regressors (excluding the constant this time) you had in the model.

The rejection / non-rejection rule would be given by selecting the appropriate region from the following diagram:


(c) We have 60 observations, and the number of regressors excluding the constant term is 3. The appropriate lower and upper limits are 1.48 and 1.69 respectively, so the Durbin Watson is lower than the lower limit. It is thus clear that we reject the null hypothesis of no autocorrelation. So it looks like the residuals are positively autocorrelated.

(d)

The problem with a model entirely in first differences, is that once we calculate the long run solution, all the first difference terms drop out (as in the long run we assume that the values of all variables have converged on their own long run values so that yt = yt-1 etc.) Thus when we try to calculate the long run solution to this model, we cannot do it because there isn’t a long run solution to this model!

(e)

The answer is yes, there is no reason why we cannot use Durbin Watson in this case. You may have said no here because there are lagged values of the regressors (the x variables) variables in the regression. In fact this would be wrong since there are no lags of the DEPENDENT (y) variable and hence DW can still be used.

6.

The major steps involved in calculating the long run solution are to

- set the disturbance term equal to its expected value of zero

- drop the time subscripts

- remove all difference terms altogether since these will all be zero by the definition of the long run in this context.

Following these steps, we obtain


We now want to rearrange this to have all the terms in x2 together and so that y is the subject of the formula:

The last equation above is the long run solution.

7. Ramsey’s RESET test is a test of whether the functional form of the regression is appropriate. In other words, we test whether the relationship between the dependent variable and the independent variables really should be linear or whether a non-linear form would be more appropriate. The test works by adding powers of the fitted values from the regression into a second regression. If the appropriate model was a linear one, then the powers of the fitted values would not be significant in this second regression.

If we fail Ramsey’s RESET test, then the easiest “solution” is probably to transform all of the variables into logarithms. This has the effect of turning a multiplicative model into an additive one.

If this still fails, then we really have to admit that the relationship between the dependent variable and the independent variables was probably not linear after all so that we have to either estimate a non-linear model for the data (which is beyond the scope of this course) or we have to go back to the drawing board and run a different regression containing different variables.

8. (a) It is important to note that we did not need to assume normality in order to derive the sample estimates of and or in calculating their standard errors. We needed the normality assumption at the later stage when we come to test hypotheses about the regression coefficients, either singly or jointly, so that the test statistics we calculate would indeed have the distribution (t or F) that we said they would.

(b) One solution would be to use a technique for estimation and inference which did not require normality. But these techniques are often highly complex and also their properties are not so well understood, so we do not know with such certainty how well the methods will perform in different circumstances.


One pragmatic approach to failing the normality test is to plot the estimated residuals of the model, and look for one or more very extreme outliers. These would be residuals that are much “bigger” (either very big and positive, or very big and negative) than the rest. It is, fortunately for us, often the case that one or two very extreme outliers will cause a violation of the normality assumption. The reason that one or two extreme outliers can cause a violation of the normality assumption is that they would lead the (absolute value of the) skewness and / or kurtosis estimates to be very large.

Once we spot a few extreme residuals, we should look at the dates when these outliers occurred. If we have a good theoretical reason for doing so, we can add in separate dummy variables for big outliers caused by, for example, wars, changes of government, stock market crashes, changes in market microstructure (e.g. the “big bang” of 1986). The effect of the dummy variable is exactly the same as if we had removed the observation from the sample altogether and estimated the regression on the remainder. If we only remove observations in this way, then we make sure that we do not lose any useful pieces of information represented by sample points.

9. (a) Parameter structural stability refers to whether the coefficient estimates for a regression equation are stable over time. If the regression is not structurally stable, it implies that the coefficient estimates would be different for some sub-samples of the data compared to others. This is clearly not what we want to find since when we estimate a regression, we are implicitly assuming that the regression parameters are constant over the entire sample period under consideration.

(b) 1981M1-1995M12rt = 0.0215 + 1.491 rmt RSS=0.189 T=180

1981M1-1987M10rt = 0.0163 + 1.308 rmt RSS=0.079 T=82

1987M11-1995M12rt = 0.0360 + 1.613 rmt RSS=0.082 T=98

(c) If we define the coefficient estimates for the first and second halves of the sample as 1 and 1, and 2 and 2 respectively, then the null and alternative hypotheses are

H0 : 1 = 2 and 1 = 2

and H1 : 1 2 or 1 2


(d) The test statistic is calculated as

Test stat. =

This follows an F distribution with (k,T-2k) degrees of freedom. F(2,176) = 3.05 at the 5% level. Clearly we reject the null hypothesis that the coefficients are equal in the two sub-periods.

10. The data we have are1981M1-1995M12

rt = 0.0215 + 1.491 Rmt RSS=0.189 T=1801981M1-1994M12

rt = 0.0212 + 1.478 Rmt RSS=0.148 T=1681982M1-1995M12

rt = 0.0217 + 1.523 Rmt RSS=0.182 T=168

First, the forward predictive failure test - i.e. we are trying to see if the model for 1981M1-1994M12 can predict 1995M1-1995M12.The test statistic is given by

Where T1 is the number of observations in the first period (i.e. the period that we actually estimate the model over), and T2 is the number of observations we are trying to “predict”. The test statistic follows an F-distribution with (T2, T1-k) degrees of freedom. F(12, 166) = 1.81 at the 5% level. So we reject the null hypothesis that the model can predict the observations for 1995. We would conclude that our model is no use for predicting this period, and from a practical point of view, we would have to consider whether this failure is a result of a-typical behaviour of the series out-of-sample (i.e. during 1995), or whether it results from a genuine deficiency in the model.

The backward predictive failure test is a little more difficult to understand, although no more difficult to implement. The test statistic is given by


Now we need to be a little careful in our interpretation of what exactly are the “first” and “second” sample periods. It would be possible to define T1 as always being the first sample period. But I think it easier to say that T1 is always the sample over which we estimate the model (even though it now comes after the hold-out-sample). Thus T2 is still the sample that we are trying to predict, even though it comes first. You can use either notation, but you need to be clear and consistent. If you wanted to choose the other way to the one I suggest, then you would need to change the subscript 1 everywhere in the formula above so that it was 2, and change every 2 so that it was a 1.

Either way, we conclude that there is little evidence against the null hypothesis. Thus our model is able to adequately back-cast the first 12 observations of the sample.

11. By definition, variables having associated parameters that are not significantly different from zero are not, from a statistical perspective, helping to explain variations in the dependent variable about its mean value. One could therefore argue that empirically, they serve no purpose in the fitted regression model. But leaving such variables in the model will use up valuable degrees of freedom, implying that the standard errors on all of the other parameters in the regression model, will be unnecessarily higher as a result. If the number of degrees of freedom is relatively small, then saving a couple by deleting two variables with insignificant parameters could be useful. On the other hand, if the number of degrees of freedom is already very large, the impact of these additional irrelevant variables on the others is likely to be inconsequential.

12. An outlier dummy variable will take the value one for one observation in the sample and zero for all others. The Chow test involves splitting the sample into two parts. If we then try to run the regression on both the sub-parts but the model contains such an outlier dummy, then the observations on that dummy will be zero everywhere for one of the regressions. For that sub-sample, the outlier dummy would show perfect multicollinearity with the intercept and therefore the model could not be estimated.


Chapter4 Solutions

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