Solution to Homework 4

Economics 421/521
Winter 2012
Solution to Homework #4

1. Perform a Durbin-Watson test at the 5% level of significance for positive first-order autocorrelation using the following regression output (standard errors in parentheses):

Y_t = 2.0 + 3.7*X_1t - 4.4*X_2t,    T = 42
       (.7)   (1.1)          (2.8)           DW = 1.22

At the 5% level of significance, the critical value of the DW statistic is (approximately, with interpolation) 1.41 at the lower end and 1.61 at the upper end. Since 1.22 is smaller than d_L = 1.41, we reject that ρ=0.

2. Recall the model from homework 1:

Given data on M2, real GDP, and the T-bill rate, estimate the following regression...:

M_t = β₀ +  β₁RGDP_t +  β₂Tbill_t + e_t

Don't be surprised if the fit is very good - we'll explain why that may be misleading later in the course.

Here's the output:

[Click on figure for larger version]

Does model suffer from serial correlation? Use a Durbin-Watson test to answer the question.

Yes, definitely. The value of d_L (5%) is, approximately, 1.63. The test statistic of .02891 is far below this value, so we reject that ρ=0.

Is the fit as good as the R² and t-statistics indicate?

The t-statistics are biased upward so the fit is not nearly as good as the t-statistics might lead you to believe (which would be evident if we corrected for it). This is due to biased estimates of the residuals. So the the R² and t-statistics give a misleading picture of how well the model fits the data.

3. Regress the change in the log of real consumption (C) on the change in the log of real disposable income (DI) and test for serial correlation using a Durbin-Watson test. The data are here (the data are quarterly, and span the time period 1947:Q1 - 2007:Q3).

The first step is to log both consumption and disposable income, then difference. The transformed variables are named dlogc and dlogy below (e.g. dlogy = log(di)-log(di(-1)). Then, regress dlogc on a constant and dlogy:

Here is the regression output:

The critical value of the Durbin-Watson statistic is approximately d_L = 1.65 and  d_U = 1.69 (there are tables that go beyond T=100, but the value doesn't change much after 100 observations. I used the value for 100 observations from the table in the text). For a test of negative serial correlation the values are d_L = 4.00-1.65 = 2.35 and  d_U = 4.00-1.69 = 2.31. The null of no positive serial correlation would not be rejected since the test statistic, 2.28, is above 1.65. the rejection point. Similarly, a null of no negative serial correlation cannot be rejected since 2.28 is less than 2.31, but it's a close call.

4. Explain why the Durbin-Watson statistic is always between 0 and 4. Also explain why the Durbin-Watson statistic is between 0 and 2 when there is positive serial correlation, between 2 and 4 when there is negative serial correlation, and equal to 2 when there is no correlation at all.

Start with the demonstration that d, the Durbin-Watson statistic, is approximately (2-2ρ) in large samples:

[Click on figure for larger version]

Now, since ρ must lie between -1 and 1, the Durbin-Watson statistic must lie between 0 and 4 [since 2-2(1)=0 and 2-2(-1)=4].

To see the last part, note that when ρ=0, d=2. Then, as ρ increases from 0 to 1, d moves from 2 to 0, and as ρ moves from 0 to -1, d moves from 2 to 4.

5. Continuing with the model we used in problem 2, test for the presence of fourth order serial correlation.

To do this problem, first regress m2 on a constant, rgdp, and tbillrate:

[Click on figure for larger version]

Save the residuals (I saved the resid series as uhat). Regress the estimated residual on four lags of the estimated residual and the other variables in the model (no constant), i.e. regress uhat_t on uhat_t-1, uhat_t-2, uhat_t-3, uhat_t-4, tbillrate, and rgdp:

[Click on figure for larger version]

Finally, form the test statistic (T-P)R², where T is the number of observations and P is the number of lags. This is distributed χ²(4). The critical value for this test at  5% level of significance is 9.49.

The test statistic is (191)(.972914) = 185.83, so reject that there is no serial correlation in the model.

6. Continuing with the model we used in problem 3, use the AR(1) procedure in EViews to correct the model for the presence of first-order serial correlation.

First, note that the test above did not indicate the presence of serial correlation, so technically there is no need for a correction. However, it's still worthwhile to do as a practice exercise.

To do this problem, just add the ar(1) command to the estimation shown in problem 3:

With the result: