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Midterm Econ 4304, Spring 2022

Q1: (10 points) Consider the following GARCH(1,1) model:

  = 1 +   ,            = √ℎ   ,         ℎ  = 1 + 0.2 ⋅  −1  + 0.3 ⋅ ℎ−1

( |−1) = 0,   ( |−1) = 1

We observe yt . It  denotes the information available at time t.

(1)  Show that ( |−1) = 0.

(2)  Show that ( |−1) = ( |−1) = ℎ  .

(3)  Show that  (ℎ | ) =  ( | ),  ≥ 1. (Hint: use “ant eats elephant” { ( | )|} = ( | )    .)

(4)  Variance forecast: calculate (+ | ),  ≥ 1.

 

Q2: (10 points) Pick one individual stock (not a stock index) of your own choice and perform the               following analysis using R. Transform the data into monthly data series. Define the monthly return as     log-difference of adjusted closing price. You may consult the sample R scripts on Canvas while doing this empirical problem.

(1)  Provide a summary statistics table which includes the range of sample period, the mean, median,  minimum, maximum, standard deviation. Also provide a table for the autocorrelations (ACF/PACF) at lag order 1, 4, 8, and 12 and the corresponding Q-statistics with p-values.

(2)  Estimate an adequate GARCH or AR-GARCH model for the return series, depending on your findings in (1). You need to demonstrate that there is no significant serial correlations in residuals ut , the

standardized residuals vt , and vt(2) . As an example for notations, the AR(1)-GARCH(1,1) model is

Rt  =  + −1  +   ,           =    ,

ℎ  =  +   −1  +  ⋅ ℎ−1, ( |−1) = 0,   ( |−1) = 1           Please note that the adequate model is not necessarily AR(1)-GARCH(1,1). It could be any AR(k)- GARCH(p,q), etc.

(3)  Assume that vt ~(0, 1). Calculate the 10%-Value at Risk of the return series. Also calculate the 10%-Expected Shortfall of the return series. Hint: In R, the function to get quantiles of N(0,1) is qnorm(). For example, qnorm(0.025) = -1.96, qnorm(0.05)=-1.645. The function to calculate       exponentials is exp(). For example, exp(1) = 2.718, exp(-0.5)=0.6065.

(4)  Perform out-of-sample forecast for √ℎ  from March 2022 to March 2023. Use a table and a graph to report your results.


Q3: (10 points) Perform ARMA analysis of the US housing start. Use R to directly load the data code “HOUST” from the database FRED” . The source of the data is at:

https://fred.stlouisfed.org/series/HOUST

(1)  Plot the time series. Plot the ACF and PACF for 24 lags.

(2)  Estimate an AR(12) model and perform serial correlation test of its residuals to see whether they are white noise.

(3)  Find an AR model that has a smaller AIC than the benchmark AR(12) model? In the same time, your model should produce white noise residuals. Hint: If we need to estimate a restricted AR(13) model such as: yt  =  +  1 −1  +  2 −2  +  3  −3  +  12 −12  +  13 −13  +   .

In R use the script:

ar13 <- arima(y,

order = c(13,0,0),

fixed = c(NA,NA,NA,rep(0,8),NA,NA,NA))

To obtain an adequate model, you might need to try AR(24) or AR(25) model. Try to remove certain insignificant coefficients to reduce the AIC of the estimated models.

(4)  Use your chosen model from (3) to conduct out-of-sample forecast for the housing start for the next

24 months. Plot the fanchart with 60%, 80%, and 95% confidence intervals. Also provide the forecast and the confidence intervals in a table.