Econ 240 C
description
Transcript of Econ 240 C
![Page 1: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/1.jpg)
Econ 240 C
Lecture 16
![Page 2: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/2.jpg)
2
Part I. ARCH-M Modeks
In an ARCH-M model, the conditional variance is introduced into the equation for the mean as an explanatory variable.
ARCH-M is often used in financial models
![Page 3: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/3.jpg)
3Net return to an asset model Net return to an asset: y(t)
• y(t) = u(t) + e(t)• where u(t) is is the expected risk premium• e(t) is the asset specific shock
the expected risk premium: u(t)• u(t) = a + b*h(t)• h(t) is the conditional variance
Combining, we obtain:• y(t) = a + b*h(t) +e(t)
![Page 4: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/4.jpg)
4Northern Telecom And Toronto Stock Exchange
Nortel and TSE monthly rates of return on the stock and the market, respectively
Keller and Warrack, 6th ed. Xm 18-06 data file
We used a similar file for GE and S_P_Index01 last Fall in Lab 6 of Econ 240C
![Page 5: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/5.jpg)
5
![Page 6: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/6.jpg)
6Returns Generating Model, Variables Not Net of Risk Free
![Page 7: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/7.jpg)
7
![Page 8: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/8.jpg)
8Diagnostics: Correlogram of the Residuals
![Page 9: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/9.jpg)
9Diagnostics: Correlogram of Residuals Squared
![Page 10: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/10.jpg)
10
![Page 11: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/11.jpg)
11Try Estimating An ARCH-GARCH Model
![Page 12: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/12.jpg)
12
![Page 13: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/13.jpg)
13Try Adding the Conditional Variance to the Returns Model PROCS: Make GARCH variance series:
GARCH01 series
![Page 14: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/14.jpg)
14Conditional Variance Does Not Explain Nortel Return
![Page 15: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/15.jpg)
15
![Page 16: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/16.jpg)
16OLS ARCH-M
![Page 17: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/17.jpg)
17
Estimate ARCH-M Model
![Page 18: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/18.jpg)
18Estimating Arch-M in Eviews with GARCH
![Page 19: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/19.jpg)
19
Part II. Granger Causality
Granger causality is based on the notion of the past causing the present
example: Lab six, Index of Consumer Sentiment January 1978 - March 2003 and S&P500 total return, montly January 1970 - March 2003
![Page 20: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/20.jpg)
20Consumer Sentiment and SP 500 Total Return
![Page 21: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/21.jpg)
21
Time Series are Evolutionary
Take logarithms and first difference
![Page 22: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/22.jpg)
22
![Page 23: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/23.jpg)
23
![Page 24: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/24.jpg)
24
Dlncon’s dependence on its past
dlncon(t) = a + b*dlncon(t-1) + c*dlncon(t-2) + d*dlncon(t-3) + resid(t)
![Page 25: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/25.jpg)
25
![Page 26: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/26.jpg)
26Dlncon’s dependence on its past and dlnsp’s past
dlncon(t) = a + b*dlncon(t-1) + c*dlncon(t-2) + d*dlncon(t-3) + e*dlnsp(t-1) + f*dlnsp(t-2) + g* dlnsp(t-3) + resid(t)
![Page 27: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/27.jpg)
27
![Page 28: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/28.jpg)
Do lagged dlnsp terms add to the explained variance?
F3, 292 = {[ssr(eq. 1) - ssr(eq. 2)]/3}/[ssr(eq. 2)/n-7] F3, 292 = {[0.642038 - 0.575445]/3}/0.575445/292 F3, 292 = 11.26 critical value at 5% level for F(3, infinity) = 2.60
![Page 29: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/29.jpg)
29
Causality goes from dlnsp to dlncon
EVIEWS Granger Causality Test• open dlncon and dlnsp• go to VIEW menu and select Granger Causality• choose the number of lags
![Page 30: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/30.jpg)
30
![Page 31: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/31.jpg)
31Does the causality go the other way, from dlncon to dlnsp? dlnsp(t) = a + b*dlnsp(t-1) + c*dlnsp(t-2) +
d* dlnsp(t-3) + resid(t)
![Page 32: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/32.jpg)
32
![Page 33: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/33.jpg)
33Dlnsp’s dependence on its past and dlncon’s past dlnsp(t) = a + b*dlnsp(t-1) + c*dlnsp(t-2) +
d* dlnsp(t-3) + e*dlncon(t-1) + f*dlncon(t-2) + g*dlncon(t-3) + resid(t)
![Page 34: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/34.jpg)
34
![Page 35: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/35.jpg)
Do lagged dlncon terms add to the explained variance?
F3, 292 = {[ssr(eq. 1) - ssr(eq. 2)]/3}/[ssr(eq. 2)/n-7] F3, 292 = {[0.609075 - 0.606715]/3}/0.606715/292 F3, 292 = 0.379 critical value at 5% level for F(3, infinity) = 2.60
![Page 36: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/36.jpg)
36
![Page 37: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/37.jpg)
37Granger Causality and Cross-Correlation
One-way causality from dlnsp to dlncon reinforces the results inferred from the cross-correlation function
![Page 38: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/38.jpg)
38
![Page 39: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/39.jpg)
39Part III. Simultaneous Equations and Identification
Lecture 2, Section I Econ 240C Spring 2003
Sometimes in microeconomics it is possible to identify, for example, supply and demand, if there are exogenous variables that cause the curves to shift, such as weather (rainfall) for supply and income for demand
![Page 40: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/40.jpg)
40
Demand: p = a - b*q +c*y + ep
![Page 41: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/41.jpg)
41
demand
price
quantity
Dependence of price on quantity and vice versa
![Page 42: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/42.jpg)
42
demand
price
quantity
Shift in demand with increased income
![Page 43: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/43.jpg)
43
Supply: q= d + e*p + f*w + eq
![Page 44: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/44.jpg)
44
price
quantity
supply
Dependence of price on quantity and vice versa
![Page 45: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/45.jpg)
45
Simultaneity
There are two relations that show the dependence of price on quantity and vice versa• demand: p = a - b*q +c*y + ep
• supply: q= d + e*p + f*w + eq
![Page 46: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/46.jpg)
46
Endogeneity
Price and quantity are mutually determined by demand and supply, i.e. determined internal to the model, hence the name endogenous variables
income and weather are presumed determined outside the model, hence the name exogenous variables
![Page 47: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/47.jpg)
47
price
quantity
supply
Shift in supply with increased rainfall
![Page 48: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/48.jpg)
48
Identification
Suppose income is increasing but weather is staying the same
![Page 49: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/49.jpg)
49
demand
price
quantity
Shift in demand with increased income, may trace outi.e. identify or reveal the demand curve
supply
![Page 50: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/50.jpg)
50
price
quantity
Shift in demand with increased income, may trace outi.e. identify or reveal the supply curve
supply
![Page 51: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/51.jpg)
51
Identification
Suppose rainfall is increasing but income is staying the same
![Page 52: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/52.jpg)
52
price
quantity
supply
Shift in supply with increased rainfall may trace out, i.e. identify or reveal the demand curve
demand
![Page 53: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/53.jpg)
53
price
quantity
Shift in supply with increased rainfall may trace out, i.e. identify or reveal the demand curve
demand
![Page 54: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/54.jpg)
54
Identification
Suppose both income and weather are changing
![Page 55: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/55.jpg)
55
price
quantity
supply
Shift in supply with increased rainfall and shift in demandwith increased income
demand
![Page 56: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/56.jpg)
56
price
quantity
Shift in supply with increased rainfall and shift in demandwith increased income. You observe price and income
![Page 57: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/57.jpg)
57
Identification
All may not be lost, if parameters of interest such as a and b can be determined from the dependence of price on income and weather and the dependence of quantity on income and weather then the demand model can be identified and so can supply
![Page 58: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/58.jpg)
The Reduced Form for p~(y,w)
demand: p = a - b*q +c*y + ep
supply: q= d + e*p + f*w + eq
Substitute expression for q into the demand equation and solve for p
p = a - b*[d + e*p + f*w + eq] +c*y + ep
p = a - b*d - b*e*p - b*f*w - b* eq + c*y + ep
p[1 + b*e] = [a - b*d ] - b*f*w + c*y + [ep - b* eq ] divide through by [1 + b*e]
![Page 59: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/59.jpg)
The reduced form for q~y,w demand: p = a - b*q +c*y + ep
supply: q= d + e*p + f*w + eq
Substitute expression for p into the supply equation and solve for q
supply: q= d + e*[a - b*q +c*y + ep] + f*w + eq
q = d + e*a - e*b*q + e*c*y +e* ep + f*w + eq
q[1 + e*b] = [d + e*a] + e*c*y + f*w + [eq + e* ep]
divide through by [1 + e*b]
![Page 60: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/60.jpg)
60
Note: the coefficient on income, y, in the equation for q, divided by the coefficient on income in the equation for p equals e, the slope of the supply equation
Note: the coefficient on weather in the equation for for p, divided by the coefficient on weather in the equation for q equals -b, the slope of the demand equation
![Page 61: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/61.jpg)
61
From these estimates of e and b we can calculate [1 +b*e] and obtain c from the coefficient on income in the price equation and obtain f from the coefficient on weather in the quantity equation
it is possible to obtain a and d as well
![Page 62: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/62.jpg)
62
Vector Autoregression (VAR)
Simultaneity is also a problem in macro economics and is often complicated by the fact that there are not obvious exogenous variables like income and weather to save the day
As John Muir said, “everything in the universe is connected to everything else”
![Page 63: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/63.jpg)
63VAR One possibility is to take advantage of the
dependence of a macro variable on its own past and the past of other endogenous variables. That is the approach of VAR, similar to the specification of Granger Causality tests
One difficulty is identification, working back from the equations we estimate, unlike the demand and supply example above
We miss our equation specific exogenous variables, income and weather
![Page 64: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/64.jpg)
Primitive VAR(1)y(t) = w(t) + y(t-1) +w(t-1) + x(t) + ey
(t)
(2) w(t) = y(t) + y(t-1) +w(t-1) + x(t) + ew
(t)
![Page 65: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/65.jpg)
65
Standard VAR
Eliminate dependence of y(t) on contemporaneous w(t) by substituting for w(t) in equation (1) from its expression (RHS) in equation 2
![Page 66: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/66.jpg)
1. y(t) = w(t) + y(t-1) + w(t-1) + x(t) + ey(t)
1’. y(t) = y(t) + y(t-1) + w(t-1) + x(t) + ew(t)] + y(t-1) + w(t-1) + x(t) + ey(t)
1’. y(t) y(t) = [+ y(t-1) + w(t-1) + x(t) + ew(t)] + y(t-1) + w(t-1) + x(t) + ey(t)
![Page 67: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/67.jpg)
Standard VAR (1’) y(t) = (/(1- ) +[ (+ )/(1-
)] y(t-1) + [ (+ )/(1- )] w(t-1) + [(+ (1- )] x(t) + (ey(t) + ew(t))/(1- )
in the this standard VAR, y(t) depends only on lagged y(t-1) and w(t-1), called predetermined variables, i.e. determined in the past
Note: the error term in Eq. 1’, (ey(t) + ew(t))/(1- ), depends upon both the pure shock to y, ey(t) , and the pure shock to w, ew
![Page 68: Econ 240 C](https://reader035.fdocuments.us/reader035/viewer/2022081604/56814b06550346895db81d7a/html5/thumbnails/68.jpg)
Standard VAR (1’) y(t) = (/(1- ) +[ (+ )/(1-
)] y(t-1) + [ (+ )/(1- )] w(t-1) + [(+ (1- )] x(t) + (ey(t) + ew(t))/(1- )
(2’) w(t) = (/(1- ) +[(+ )/(1- )] y(t-1) + [ (+ )/(1- )] w(t-1) + [(+ (1- )] x(t) + (ey(t) + ew(t))/(1- )
Note: it is not possible to go from the standard VAR to the primitive VAR by taking ratios of estimated parameters in the standard VAR