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ECMT 6002/6702: Econometric Applications Week9 Tutorial
发布时间：2023-06-09
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ECMT 6002/6702: Econometric Applications
1 Empirical application
For the time series yt of returns on the S&P 500 index (from the“VOLAT”dataset of Wooldridge’s textbook), we will evaluate the following two prediction models:
yt = ϕ0 + ϕ1yt − 1 + et
v.s.
yt = ϕ0 + ϕ1yt − 1 + ϕ2yt −2 + et .
Instructions:
1. Compute log-returns from the S&P 500 index.
- T=length(snp);r=log(snp[2:T])-log(snp[1:(T-1)]),
where snp : S&P indices over time.
- As a result, you will obtain the time series of log-returns over t = 1, . . . , 557.
2. Estimate the prediction models with only 570 observations first and obtain 
551 . Similarly obtain 
551+k by estimating the prediction models with only 550 + k observations. Report forecasts 
551 , . . . , 
557 . .
- The result will be similar to
(i) AR(1) case :
551 = 0.010572, 
552 = 0.011859, 
553 = 0.004297, 
554 = 0.008156, 
555 = 0.009209, 
551 = 0.000653, 
557 = 0.005725.
(ii) AR(2) case :
551 = 0.012749, 
552 = 0.010975, 
553 = 0.002295, 
554 = 0.009025, 
555 = 0.008897, 
556 = −0.000796, 
557 = 0.007587.
3. Compute the mean square error (MSE) using the remaining observations y571 , . . . , y577 (note that the forecast error at time t = 570+ s is defined by 
570+s − y570+s for s = 1, . . . , 7. Which model is better?
- The results will be similar to
(i) AR(1) case : MSE=0.00020 (ii) AR(2) case : MSE=0.00019
and thus the AR(2) prediction model is slightly better.
4. Compute the mean absolute error (MAE) using the remaining observations y571 , . . . , y577 . Which model is better?
- The results will be similar to
(i) AR(1) case : MAE=0.01188 (ii) AR(2) case : MAE=0.01187
and thus the AR(2) prediction model is slightly better.
5. Compute the mean absolute percentage error (MAPE) using the remaining observations y571 , . . . , y577 . Which model is better?
- The results will be similar to
(i) AR(1) case : MAPE=257.9513 (ii) AR(2) case : MAPE=250.3959
and thus the AR(2) prediction model is slightly better.
6. Forecast comparison may be done via the so-called rolling-window approach. The only dif- ferent step for this method is 
570+k, for k = 1, . . . , 7, is obtained from the 550 recent obser- vations yk , . . . , y570+k − 1 . Repeat this excercise with the rolling-window approach.
- The results will be similar to
(i) AR(1) case : MSE=0.00020, MAE=0.01185, MAPE=259.9053. (ii) AR(2) case : MSE=0.00019, MAE=0.01181, MAPE= 249.9018.
- In this example where we compute MSE, MAE and MAPE with only 7 forecasts and real- izations, this approach will not yield significantly different results. But if MSE, MAE and
MAPE are computed from many forecasts and realizations, the results might be different.
7. This computing exercise is not mandatory.
