ECO4185 - Financial Econometrics - Summer 2022 Final exam - Part 2
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ECO4185 - Financial Econometrics - Summer 2022
Final exam - Part 2, Long Question (70 points)
In this question, we are going to use some data for Canada, which you will ind in the attached Excel ile. There are two variables in the ile. The irst variable, named “EA” and reported in Column B, is a measure of economic activity for Canada. The second variable, named “PI” and reported in Column C, is a price index. More information about each variable is included in the Excel ile.
You can choose the sample period that you prefer, but make sure that it is long enough. Since we are going to work with a bi-variate model, the sample period must be the same for both variables.
a. (15 points) Construct your dataset and save it on an Excel ile that you are asked to submit together with your answers.
i. Compute the following variables:
• yt = EAt, where EAt is your economic activity variable (Column B) for time t (we are going to use this variable as it is, with no transformations);
• dt = 100 × ln(PIt), where “ln” denotes the natural logarithm and PIt is your price index variable (Column C) for time t.
ii. Before starting with our analysis, we are going to examine whether dt contains a unit root by running the Dickey-Fuller test, augmented with one lag of the dependent variable. Estimate the following regression using OLS:
dt = c0 + ψdt−1 + c1 dt−1 + et (1)
Report your ψ六 and SE(ψ六). Write the statement of the Dickey-Fuller test in terms of the parameters of (1). Make sure to specify both the null and the alternative hypothesis. Then use your estimates to compute the Dickey-Fuller test statistic. Use the appropriate critical value for the Dickey-Fuller distribution from the table given below. Can you reject the H0 at the 5% signiicance level? Explain your answer.
iii. Our goal is to have a stationary variable xt that we can use for our analysis. Based on the results of the Dickey-Fuller test that you run, decide whether you want to set xt = dt or xt = dt − dt−1 . Explain your choice and report xt on your Excel ile. Note: it is not necessary to run the Dickey-Fuller test twice to examine whether your variable dt is potentially I(2).
iv. Finally, create the variables: yt−1 , yt−2 , xt−1, and xt−2; report them on your Excel ile. State your inal sample period (irst month & year - last month & year) and its length T (the number of observations) once you have accounted for the lags of the variables.
b. (9 points) We are interested in estimating the following equation for xt:
xt = α0 + α 1yt + α2yt−1 + α3yt−2 + e1,t (2)
where e1,t is a Normally distributed i.i.d. white noise process with variance ae(2)1 . We think that (2) satisies all the other assumptions that we need to be able to estimate its parameters by OLS.
Our main goal is to obtain a reliable estimate of the parameter α 1 , which measures the contemporaneous efect of yt on xt .
i. Estimate (2) by OLS and report your estimated parameters and standard errors (it is enough to mark them in the ile with the codes/results from your statistical software).
ii. Use your estimated model to predict the change in xt following an exogenous increase in yt by 0.25 percentage points. Provide a 95% conidence interval for this change; you can use the critical values for the N (0, 1) distribution to compute this interval. Show your work.
Then interpret your result in terms of the original variables EAt and PIt: how is PIt predicted to change if EAt exogenously increases by 0.25 percentage points? Explain your answer.
c. (12 points) We decide to consider the possibility of a feedback efect from xt to yt . The speciic equation that we have in mind for yt is:
yt = β0 + β1 xt + β2 yt−1 + β3 xt−1 + e2,t (3)
where e2,t is a Normally distributed i.i.d. white noise process with variance ae(2)2 . We think that it is reasonable to assume that E(e1,te2,t−k) = 0 for all k .
i. Write the reduced-form equations for xt and yt originating from the bi-variate structural model composed of (2) and (3). Show your work.
Then use the reduced-form equation for yt to write the expression for E(yte1,t). Show that if the parameter β 1 in (3) is diferent from zero, then the assumption E(yte1,t) = 0 does not hold in the bi-variate structural model.
ii. We want to examine the evidence that β 1 0 in equation (3). Our friend Paula suggests estimating (3) by OLS and then testing the null hypothesis H0 : β 1 = 0 with a two-sided alternative hypothesis. She says that if we reject the H0 of this test, then we do have evidence that β 1 0.
Do you think this is an appropriate strategy for our goal of inferring whether β 1 0? Why or why not? Make sure to clearly motivate your answer; points will be given only on the basis of the explanation provided.
Note: the question is not asking you to actually run the test.
iii. Again, we want to investigate the evidence that β1 0 in equation (3). Our friend Omar suggests using a Granger-causality test for this purpose. Speciically, he suggests estimating the following reduced-form VAR model:
Zt = B0 + B1Zt−1 + B2Zt−2 + ut (4)
where Zt = [xt yt]′ , ut is a (2 × 1) vector of reduced-form innovations, and B0 , B1, and B2 contain parameters.
Omar says that we should then test the H0 that xt does not Granger-cause yt . He argues that if we reject the H0 of this test, then we do have evidence that β 1 0 in (3).
Do you think this is an appropriate strategy for our goal of inferring whether β 1 0? Why or why not? Again, make sure to clearly explain your answer; points will be given only on the basis of the explanation provided.
Note: again, the question is not asking you to actually run the test. Hint: I think it may be helpful to write the statement of the Granger-causality test in terms of the parameters of (4), and then compare the restricted and unrestricted VAR models involved in the test with the reduced form equations for xt and yt originating from (2) and (3).
d. (12 points) Consider again the bi-variate model composed of (2) and (3). We are still interested in obtaining a reliable estimate of the parameter α 1 , and we want to use the IV method for this purpose. Speciically, we plan to replace yt with the variable t and estimate the following version on equation (2):
xt = α0 + α 1 t + α2 yt−1 + α3 yt−2 + w1,t (5)
i. We plan to compute t as the itted values of a linear regression model estimated using OLS. Speciically, I suggest using the following model:
yt = c0 + c1yt−1 + c2 yt−2 + w2,t (6)
Do you think that the variable t computed from (6) would allow us to obtain a consistent estimate of the parameter α 1 in (5)? If your answer is yes, then estimate (6) by OLS and compute t . If your answer is no, explain what the problem is and suggest an alternative linear regression model that we can use instead of (6). Write the equation of this alternative linear regression model, then estimate its parameters using OLS and compute t . Report the variable t computed with either (6) or your alternative model in the Excel ile with your data.
Finally, estimate (5) by OLS and report your estimated parameters and standard errors (again, it is enough to mark them in the ile with the codes/results from your statistical software).
ii. Using your estimated model (5), predict again the change in xt following an exogenous increase in yt by 0.25 percentage points. Show your work. Then interpret your result in terms of your original variables EAt and PIt .
e. (10 points) We decide to perform some further analysis of the relationship between xt and yt using their reduced-form equations. In part c. of the question you should have found that the bi-variate model composed of (2) and (3) generates reduced-form equations for xt and yt in the form:
where v1,t and v2,t are the reduced-form innovations.
i. Estimate (7) and (8) using OLS. Report your estimated parameters and standard errors (once more, it is enough to show them in the ile with the codes/results from your statistical software).
ii. Let T be the last period in the sample that you have chosen. Used the estimated reduced-form equations and the actual data for T and (T − 1) to compute the forecasts T+1|T and T+1|T . Show your work. Then interpret your forecasts in terms of the original variables variables EAt and PIt .
f. (12 points) Last, we are interested in comparing the reduced-form bivariate model that we obtained
from (2) and (3) to a reduced-form VAR for the variables xt and yt . The reduced-form VAR is written as:
Zt = B0 + B1Zt−1 + B2Zt−2 + ut (9)
where Zt = [xt yt]′ , ut is a (2 × 1) vector of innovations, and B0 , B1, and B2 contain parameters. We decide to compare the two models using the method of the restricted and unrestricted model, in its multivariate version. Notice that equations (7) and (8) are just a restricted version of (9). So we treat the reduced-form model composed of (7) and (8) as our “restricted” model and the VAR (9) as our “unrestricted” model. We will use a Likelihood ratio test to contrast the restrictions on the lag structure imposed by our reduced-form model to the unrestricted VAR.
i. Write the statement of the test in terms of the parameters of (9). Explain why these are the same restrictions on the lag structure that are imposed in (7) and (8).
ii. You have already estimated the restricted model in part e. Use your estimated parameters to compute the residuals 1,t and 2,t, then use these residuals to obtain the estimated covariance matrix 六re . Report your 六re .
Repeat the procedure for the VAR model (9). Estimate the equations of the model and compute the residuals 1,t and 2,t . Use them to obtain the covariance matrix 六un . Report your 六un .
iii. Finally, compute the Likelihood Ratio: LR = T [ln(| 六re |) − ln(| 六un |)].
State the number of restrictions that are tested (m), and select the appropriate critical value for the χm(2) distribution from the table given below. Use a 5% signiicance level. Decide whether or not you can reject the H0 . Then interpret the results of your test: do your data suggest that the unrestricted VAR should be preferred to the reduced-form version of our bi-variate model? Explain your answer.
Critical values, N (0, 1) distribution
|
Signihcance level |
|
|
5% |
1% |
Critical value |
1.96 |
2.575 |
Critical values, χm(2) distribution
|
Signihcance level |
|
Degrees of freedom m |
5% |
1% |
2 |
5.991 |
9.210 |
3 |
7.815 |
11.345 |
4 |
9.488 |
13.277 |
Critical values, Dickey-Fuller distribution
|
Signihcance level |
|
Model |
5% |
1% |
no constant and no trend |
-1.941 |
-2.567 |
constant but no trend |
-2.863 |
-3.434 |
constant and trend |
-3.413 |
-3.963 |
2022-08-26