Reference no: EM132603147

MFE 6220 Quantitative Analysis in Equity Markets - Ohio University

Question 1
Mark models the monthly Mexican stock returns series (in percentages), Yt, as an AR(1) process and obtains the following results. The standard errors are given in parentheses. You can safely assume that the underlying error process for the true model is in fact a white noise error process.

Y^{^}t = 4.115 - 0.681 Y_t-1
          (2.003)   (0.292)

R² =0.64; n = 200

a. What is the necessary condition for the Yt series to be mean stationary? Conduct a suitable hypothesis test to determine whether this condition is met or not.

b. Determine the mean-reverting level for this data process.

c. Suppose the true underlying data generating process for the stock returns series is in fact the AR(1) process shown on page 1. Compute the autocorrelation coefficients up to 3 lags, and then draw the plot of the AutoCorrelation Function (ACF) up to 3 lags.

Question 2
Ana models the daily Belgian stock returns series (in percentages), Yt, as an MA(2) process and obtains the following estimation results. The standard errors are given in parentheses. You can safely assume that the underlying error process for the true model is in fact a white noise error process.

Y^{^}t = 0.821 + 0.971 e_t-1 + 1.304 e_t-2
         (0.581)  (0.133)       (0.107)

R²=0.53; n = 200

a. Determine the mean-reverting level for this data process.

b. Suppose the process is currently at . Then, in time period t a shock occurs to the system. In other words, . For simplicity, assume that there are no other shocks prior to or after this shock. Write the relevant equations to trace the impact of this shock from period (t-1) to (t+3) and show that the impact of the shock leaves the system after a finite number of periods.

c. Using your work from part (b), draw a relevant graph to trace the impact of the shock to the system. In particular, show what happens from period (t-1) to (t+3) on your graph.

d. The researcher finds out from his advisor that, twenty years ago, the true underlying data generating process for the Belgian stock returns series was in fact an MA(1) process of the following form (where the standard errors are given in parentheses):

Y^{^}t = 0.742 + 1.175 e_t-1
         (0.641)   (0.495)

R² = 0.63; n = 200

Using the formulas derived in class, compute the autocorrelation coefficients up to 3 lags, and then draw a plot of the AutoCorrelation Function (ACF) up to 3 lags.

Question 3
The SAS regression output for an AR(2)-TGARCH(1,1) model is given below for the Chinese stock market return data. Answer the questions below using this SAS output.

‘Estimation results for the AR(2)-TGARCH(1,1)-in-Mean model'

The AUTOREG Procedure
Dependent Variable: chinaret

Threshold GARCH Estimates
SSE	0.16165748	Observations	1500
MSE	0.0001078	Uncond Var	.
Log Likelihood	4825.87984	Total R-Square	0.3526
SBC	-9593.2539	AIC	-9635.7597
MAE	0.00752851	AICC	-9635.6631
MAPE	235.223423	HQC	-9619.9247

Parameter Estimates
Variable	DF	Estimate	Standard Error	t Value	Approx Pr > |t|
Intercept	1	0.0009	0.0002	3.57	0.0004
AR1	1	-0.0535	0.0268	-1.99	0.0462
AR2	1	0.0363	0.0255	1.42	0.1552
TARCHA0	1	0.2718	0.0659	4.12	<.0001
TARCHA1	1	0.0400	0.0126	3.18	0.0015
TARCHB1	1	-0.0774	0.0171	-4.52	0.0048
TGARCH1	1	0.8959	0.0155	57.78	0.0022

a. Based on the SAS output, is there evidence of a significant ARCH effect? Write down the relevant p-value and answer this question using the p-value method.

b. Is there evidence of a significant leverage effect for the Chinese stock market return data? Write down the relevant p-value and answer this question using the p-value method. If significant, then also discuss the coefficient estimate on the leverage effect variable.

c. Is the TGARCH model stationary? In order to answer this question, a relevant hypothesis test was conducted. The resulting SAS output with the test statistic and p-value are given below. Based on this output, do you conclude that the model is stationary or not? Explain clearly.

Test STATIONARITY
Source	DF	F Value	Pr > F
Numerator	1	12.58	0.0160
Denominator	1492

Use the SAS software to answer the following two questions. Remember to show your SAS program and the relevant SAS output.

Please write one complete program for each for question. So, there should be two complete SAS programs in total.

Use a 5% significance level unless otherwise stated. Use only the p-value method to answer any hypothesis test questions.

Question 4
You are given data for daily transfers over BankWire, a wire transfer system in a country that is responsible for much of the world's finances, for a recent span of 200 business days from 03/29/2010 to 12/31/2010. Import the Excel data file, named q4, into SAS and answer the following questions:

a. First, estimate a pure ARMA(1,2) model and show the estimation output. Then, estimate a pure ARMA(2,1) model and show the estimation output.
b. For each estimation from part (a), obtain the forecasted values for wire transfers for the last 15 days of the sample from 12/13/2010 to 12/31/2010. Then for each model, plot in one graph the actual and the forecasted wire transfers (along with the 95% confidence interval for the forecasts) only for the forecast horizon.

c. We often consider the optimum forecast to be that forecast which has the minimum root mean-square forecast error (RMSE), where the mean-square forecast error is the average of the sum of squared deviations between the actual and forecasted values of the data series. Compute the RMSE value for the two models and determine which model provides the better forecasts for wire transfers.

Question 5
One of the main applications of the simple linear regression model is the market model in finance. This model assumes that the rate of return on a stock is linearly related to the rate of return on the overall market. The mathematical description of the model is given below.

R_it = β₁ + β₂ R_mt + ε_t(1)

where Ri is the return on a particular stock i and Rm is the return on some composite stock market index.

You are given daily stock return data from 07/01/2015 to 06/30/2017 for Home Depot, Inc. and the S&P 500 index. Import the Excel data file, named q5, into SAS and answer the following questions:

a. Is there informal evidence of time-varying heteroskedasticity in the model? In order to answer this question, obtain a line plot of the Home Depot, Inc. stock returns.

b. Estimate the market model as an AR(2)-GARCH(1,2) estimation (don't forget to add the market return as an explanatory variable as well). Show the estimation output for this model.

c. Conduct a formal test in SAS to determine whether the AR(2)-GARCH(1,2) model you estimated is in fact stationary. Remember to show the SAS output for your hypothesis test and then determine whether the model is stationary or not.

d. For the model you estimated in part (b), obtain a line plot of the conditional variance (i.e. return volatility) estimates. Based on the line plot, when did the company experience the highest volatility in its stock return?