ECON 416/516 – Elias
Lab 4 – Serial Correlation and Heteroskedasticity in Time Series Regressions 145 points total
1. (15 points) Previously, we explained the general fertility rate, gfr (births per 1000 women of child-bearing age) in terms of the value of the personal tax exemption, pe. Specifically, we estimated this model in first differences:
∆gfrt = γ0 + δ0∆pet + δ1∆pet−1 + δ2∆pet−2 + ut
Use the data in “fertil3.RData” to run the t test for AR(1) serial correlation in the
errors, assuming strictly exogenous regressors. Briefly explain the results.
2. The file “fish.RData” contains 97 daily price and quantity observations on fish prices at the Fulton Fish Market in New York City. Use the variable ln(avgprc) which is the natural logarithm of the average daily price of fish, as the dependent variable.
(a) (20 points) Regress ln(avgprct) on four daily dummy variables, with Friday as the base. Include a linear time trend. That is, estimate the model
lnavgprct =β0 +β1mon+β2tues+β3wed+β4thurs+t+ut (1)
Use an F test at the 5% level to test the joint significance of the coefficients on the dummy variables to determine if there is evidence that price varies systematically within a week.
(b) (15 points) The variables wave2 and wave3 are measures of wave heights over the previous several days. Now, add the variables wave2 and wave3 to the regression with the four dummy variables and the time trend (i.e., model (1)). Are the variables wave2 and wave3 individually significant at the 5% level? Describe a mechanism by which stormy seas would increase the price of fish.
(c) (5 points) Compared to the estimation of model (1), what happened to the time trend when wave2 and wave3 were added to the regression? What do you suppose is going on?
(d) (5 points) Explain why all explanatory variables in the regression are safely as- sumed to be strictly exogenous.
(e) (10 points) In the regression from part (b) (i.e., the model with the daily dummy variables, the time trend, and the the two measures of wave heights), test the errors for AR(1) serial correlation, assuming strictly exogenous regressors.
(f) (15 points) In the regression from part (b) (i.e., the model with the daily dummy variables, the time trend, and the the two measures of wave heights), obtain the Newey-West standard errors using g = 4 (i.e., 4 lags). For wave2 and wave3, compare the Newey-West standard errors to the OLS standard errors that you obtained in the regression from part (b). Which standard errors did you expect to be larger?
(g) (15 points) Now, obtain the Prais-Winsten estimates for the model estimated in part (b). Are wave2 and wave3 individually statistically significant?
3. Use the data in “nyse.RData” to answer these questions.
(a) (20 points) Recall the model of stock returns previously studied,
returnt = β0 + β1returnt−1 + ut
Estimate the model and obtain the squared OLS residuals, uˆ2t . Find the average, minimum, and maximum values of uˆ2t over the sample. (The necessary R functions are “mean”, “min”, and “max”.)
(b) (10 points) Use the squared OLS residuals from the estimated model in part (a) to estimate the following model of heteroskedasticity:
Var(ut | returnt−1, returnt−2, . . .) = Var(ut | returnt−1) = δ0 + δ1returnt−1 + δ2return2t−1 That is, estimate the regression model
u2t = δ0 + δ1returnt−1 + δ2return2t−1 + vt
Report the estimated coefficients, the standard errors, the R-squared, and the ad-
justed R-squared.
(c) (5 points) Plot the conditional variance as a function of returnt−1 (i.e. create a scatterplot with returnt−1 on the horizontal axis and the fitted values from the regression in part (b) on the vertical axis). By just looking at the plot, for what value of returnt−1 is the conditional variance the smallest, and what is the variance? Approximate answers are fine. (Hint: The expression “xxx$fitted.values”, where “xxx” is the name of the linear model object you created when you ran the regression in part (b), will return the fitted value from the regression).
(d) (5 points) For predicting the dynamic variance, does the model in part (b) produce any negative variance estimates? The answer to this can be obtained from your graph in part (c), or by examining the fitted values of the regression in part (b).
(e) (5 points) Does the model in part (b) seem to fit better or worse than the first order ARCH model of stock returns that we covered in class? Explain using the adjusted R-squared for both models.
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