# Booth School of Business, University of Chicago Business 41202, Spring Quarter 2015, Mr. Ruey S. Tsay. Solutions to Midterm

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5 Answer: r t = a t, a t = σ t ɛ t with ɛ t being iid t The volatility equation is σ 2 t = ( N t 1 )a 2 t σ 2 t 1, where N t 1 = 1 if and only if a t 1 < (2 points) Based on the fitted TGARCH model m7, is the leverage effect statistically significant? Answer: Yes, the t-ratio is with p-value (2 points) Among models { m7, m4, m2 }, which one is preferred? Answer: Model m7, because it has the smallest AIC criterion. Problem C. (12 points) The consumption of natural gas in northern American cities depends heavily on temperature and daily activities. Let y be the daily sendout of natural gas and DHD be the degrees of heating days defined as DHD = 65 o F minus daily average temperature. Also, let x 1 = DHD, x 2 = lag-1 DHD, x 3 = windspeed, and x 4 be the indicator variable for weekend. The data were collected for 63 days. Statistical analysis is included in the attached R output. Answer the following questions. 1. (2 points) A multiple linear regression is applied. Write down the fitted model, including the R 2. Answer: The multiple linear regression is y = x x x x 4 +e, σ = 18.32, R 2 = (3 points) Model checking shows that the residuals have serial correlations, and the AIC selects an AR(8) model. After removing insignificant parameters, we have the model n5. Write down the fitted model. Is the model adequate? Answer: The model is (1.536B.368B 7.261B 8 )(y t 5.767x 1t 1.475x 2t 1.319x 3t x 4t ) = a t, where σ 2 = Model checking shows that the residuals have no serial correlations with Q(10) = 7.28 with p-value The model is adequate. 3. (3 points) Let φ i be the lag-i AR coefficient. It is seen that φ 1 φ 7 0.2, which is not far away from φ 8, especially in view of its standard error. This is indicative of a multiplicative model. Therefore, we fit a seasonal model. Denoted by n6. Write down the fitted seasonal model. Is the model adequate? 5

6 Answer: The model is ( B)( B 7 )(y t 5.765x 1t 1.473x 2t 1.282x 3t x 4t ) = a t, where σ 2 = The model is also adequate as its residuals have no serial correlations; see Q(10) = with p-value (2 points) Compare models n5 and n6. Which one is preferred? Answer: For in-sample fit, model n6 is preferred as it has lower AIC value. 5. (2 points) The weekend effect is rather significant in the multiple linear regression model of Question 1, but it is not so in the seasonal model. Answer: Part of the weekend effects belong to the seasonality as the period is 7. Problem D. (13 points) Consider the monthly log return of Decile 10 portfolio of CRSP from 1961,1 to with T = 648. The returns include dividends. Let r t denote the monthly log return. Answer the following questions based on the attached R output. 1. (2 points) The ACF of r t shows ˆρ 1 = and ˆρ 12 = Test H 0 : ρ 12 = 0 versus H a : ρ Compute the test statistic and draw the conclusion. Answer: t = /648 = 3.23, which is greater than Therefore, ρ 12 is not zero. 2. (2 points) The ˆρ 12 is likely due to the January effect. To remove ρ 12, one can use a simple linear regression with January dummy variable. The fitted model is r t = Jan t + ɛ t. Let r t = r t Jan t be the adjusted log returns of Decile 10 portfolio. Several models were entertained for r t ; see model g1, g2, g3 and g4. Which model is preferred? Answer: Models g3 and g4 are identical as expected. They are preferred over g1 and g2 based on the AIC criterion. 3. (2 points) Write down the fitted model g4. Answer: r t = a t a t 1, a t = σ t ɛ t with ɛ t being iid t The volatility equation is σ 2 t = ( a t a t 1 ) σ 2 t 1. 6

7 4. (2 points) Based on the fitted model g4. Is the leverage effect significant? Answer: Yes, the t-ratio is with p-value (3 points) The average of fitted volatility is and the 1% quantile of the residuals of model g4 is Compute the ratio σ2 t ( 0.171) of σt 2(0.171) model g4, where σt 2 (a t ) denotes the conditional variance of the series when innovation is a t. Answer: σ 2 t (.171) σ 2 t (.171) = ( (.171)) (0.0614) ( (.171)) (.0614) 2 = (2 points) Consider the forecasts of model g4. Why is the 1-step ahead forecast of r t different from multi-step ahead forecasts? Answer: Because of the MA(1) model of the mean equation. 7

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