PRACTICE
FINAL EXAM
INSTRUCTIONS: Answer 10 questions. Each is worth 10 points and
should take less than 10 minutes.
1. If you estimate a GJR-GARCH model and get parameters
2
2
2
𝜎𝑡2 = .000006 + .02𝑟𝑡−1
+ .05𝑟𝑡−1
𝐼(𝑟𝑡−1 <0) +. 94𝜎𝑡−1
Find a forecast of tomorrow’s volatility if today it was 30% and the
stock went up by 1%.____________
.29=sqrt(.000006+.02*.01^2 +0+.94*.3^2/252)*sqrt(252)
2. Why it is in general important to test for non-stationarity in time
series data before attempting to build an empirical model?
See Module 3: (1) non-stationary time series can’t be forecasted long term, (2) if non-stationary X
is regressed on a non-stationary Y we may get a spurious regression (explain why). Explain when
we do not get a spurious regression (cointegration).
3. The GJR-GARCH model for AAPL has the following statistics:
GARCH Model
Distribution
: gjrGARCH(1,1)
: norm
Optimal Parameters
-----------------------------------Estimate Std. Error t value Pr(>|t|)
omega
0.000024
0.000004
6.1741 0.00000
alpha1 0.011160
0.010268
1.0869 0.27708
1
beta1
gamma1
0.804615
0.222375
0.025453
0.037136
31.6118
5.9881
0.00000
0.00000
a. Is the effect of a positive return significantly different from a
negative return? Explain.
Yes, because the t-statistic on the asymmetric term gamma1 has a p-value almost zero. Since it is
less that 5% or even 1% it is statistically significant.
b. Is the effect of a positive return significantly different from
zero? Explain.
No, because the t-statistic on the ARCH term has a p-value of .28>5%. Thus ARCH term (the
effect of positive return) is not significant.
4. If the standardized residuals from GARCH estimation have a 1%
quantile of -3.0, and the volatility of this asset is forecast to be 60%,
find the 1% one day Value at Risk of a portfolio with $500,000 of
this asset.
$56,700 = 3.0*.6/sqrt(252) *500000
Multiplier*sig*Value
5. Assuming Normal distribution for returns and the volatility forecast
of 40%, find the 1% ten-day Value at Risk of a portfolio with
$500,000 of this asset (assume that volatility is constant for 10 days
and returns are independent).
$92,829.42=sqrt(10)*2.33*.4/sqrt(252)*500000
6. Use the daily data from AAPL below
Price=AAPL adjusted close price
ret = the log return
(a) Discuss stationarity/nonstationarity behaviour of the two series from
their graphs and ACF functions.
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Autocorrelations of series ‘price’, by lag
1
2
3
4
5
6
7
8
9
10
0.998 0.995 0.993 0.990 0.988 0.985 0.983 0.980 0.978 0.976
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Autocorrelations of series ‘ret’, by lag
1
0.027
2
3
0.007 -0.035
4
5
6
0.025 -0.025 -0.001
7
8
9
0.006 -0.057 -0.001
10
0.017
See discussions of stationarity/non-stationarity from Module 3 lecture.
(b) The Augmented Dickey Fuller (ADF) tests for AAPL price and
returns are reported below. Test unit root (nonstationarity)
hypothesis for each series.
library(tseries)
adf.test(price_AAPL)
Augmented Dickey-Fuller Test
data: price_AAPL
Dickey-Fuller = -1.2358, Lag order = 10, p-value = 0.9007
alternative hypothesis: stationary
adf.test(ret_AAPL)
Augmented Dickey-Fuller Test
data: ret_AAPL
Dickey-Fuller = -10.598, Lag order = 10, p-value = 0.01
alternative hypothesis: stationary
Price is nonstationary since the p-value for the unit root test is much higher than 5%. Returns
are stationary since p-value is much smaller than 5%.
7. Suppose you had data on stock price indices in two different
countries. You are interested in causality issues and want to find out
whether movements in stock prices in one country effect stock prices in
another. Describe all the steps you would go through (i.e. what models
you would estimate and what testing procedures you would use) to
investigate your question of interest.
1.Test for stationarity each series X and Y. Perform ADF test.
2. If X and Y are nonstationary test for cointeration. Engle-Granger test: Regress Y on X and
Constant C and test whether the residual is stationary. If the residual e=Y-c-bX is stationary there
is cointegrating relationship between X and Y.
3. Estimate Vector Autoregression VAR or Vector Error Correction VECM depending on whether
you found nonstationarity and cointegration.
-If Y and X are stationary use X and Y in VAR. Regress each variable on its own past lags and lags
of all other variables.
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-If Y and X nonstationary and NOT cointegrated use VAR for ∆Y=change in Y and ∆X=change in
X assuming that the differenced variables are stationary.
-If you found cointegration use VECM. VAR model+ error correction. Error from cointegration
equation with lag in each equation of the VAR model.
Use cointegrating term e(t-1), ∆Y and ∆X in VECM.
4. Test for Granger causality: Does X Granger cause Y? Does Y Granger cause X?
The coefficients of lagged variables X in equation for Y should be tested for significance in order to
find out whether X granger causes Y and vice versa.
5. Do impulse response analysis forecasting if shock in X has future effect on Y and vice versa
using e.g. 10 step forecast.
6. Perform forecasting for each variable in the model
8. The results of the probit model are given below. Explain each
variable having a positive or negative effect on probability of
bankruptcy.
tdta = "Debt to Assets"
gempl = "Employee Growth Rate"
opita = "Operating Income to Assets"
invsls = "Inventory to Sales"
lsls = "Log of Sales"
probit
(Intercept)
tdta
gempl
opita
invsls
lsls
Estimate Std. Error
-0.986683
0.654128
2.989447
0.653915
-3.768789
1.021963
-2.638085
1.572997
2.236799
1.318619
-0.135258
0.073346
z value
-1.5084
4.5716
-3.6878
-1.6771
1.6963
-1.8441
Pr(>|z|)
0.1314539
4.84e-06
0.0002262
0.0935214
0.0898255
0.0651694
***
***
.
.
.
Answer: For example, tdta has a positive significant coefficient which implies that controlling for
changes in other four variables in the probit model, a higher total debt-to-total assets ratio
increases the probability that a firm will be in distress next year (or lowers probability that the firm
will be classified as healthy). The positive relationship between financial leverage (short-term and
long-term debt) and financial distress is expected since financial leverage implies mandatory
payments of principal and interest. Firms that have high financial leverage are more likely to fail
when their income or cash flow is low.
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9. Test the null hypothesis that IPO and SEO firms are not different in
terms of their debt ratio using a t-test with t-stat=8.48. IPO firms
statistics are: mean=.2, stdev=.3, Nobs=200. SEO firms statistics
are: mean=.5, stdev=.4, Nobs=200.
Answer: the t-test=8.48 is much bigger than 2 (which is approximately 95% critical value),
and the p-value (if estimated) would be less than 5%. Thus, we reject the null hypothesis
and find that the difference in the means of two groups is statistically significant. Thus,
SEO firms have on average higher debt than IPO firms based on reported higher mean for
SEO firms.
10. The results of the estimated CAPM and Multifactor models for
the returns of the GLCGX fund (Large Cap Growth) are given
below.
(a) What is the implication of intercept coefficient? What is the
interpretation of each slope coefficient ?
Slope coefficients are factor sensitivities or exposures: 𝛽𝑖,𝑚𝑎𝑟𝑘𝑒𝑡 , 𝛽𝑖,𝑠𝑖𝑧𝑒 , 𝛽𝑖,𝑉𝑎𝑙𝑢𝑒
and 𝛽𝑖,𝑚𝑜𝑚 . Intercept 𝛼𝑖 shows the performance of investment i against CAPM or
Fama-French model in expectations. For example, 𝛽𝑖,𝑚𝑎𝑟𝑘𝑒𝑡 indicates that a 1%
rise or fall in the market results in a 𝛽𝑖,𝑚𝑎𝑟𝑘𝑒𝑡 × 1% rise or fall
in the return of security i. Note that GLCGX fund’s exposure to SMB (small minus
big) and HML (high minus low book to market) factors have negative betas. This
is as expected since the GLCGX fund is a large cap growth fund, thus, its
exposure to small stocks and value stocks is negative.
Performance test (Jensen’s alpha). If the t-test from the regression indicates that
𝛼 > 0 and statistically significant it would mean that the asset is yielding an
excess return higher than the CAPM or Fama-French model predicts. In the
results below for the CAPM model p-value for the intercept t-test is less than 0.05
but in Fama French model p-value is above 0.05. Thus, based on the benchmark
CAPM model GLCGX fund underperformed. Based on Fama-French multifactor
model it performed as expected.
(b) For each regression, what is the proportion of systematic risk
and what is the proportion of firm specific, idiosyncratic or
unsystematic risk?
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For the CAPM regression the R-squared=0.87. Thus, 87% of risk is market
systematic risk. The rest 13% is unsystematic or idiosyncratic risk. For the
Multifactor Model R-squared=0.93 and 93% of risk is factors systemic risk. The
multifactor model explains larger proportion of GLCGX risk.
(c) Which model better explains GLCGX fund returns: the CAPM or
the Multifactor Model?
Based on Adjusted R-squared= 0.9297 we find that the multifactor model has
better statistical performance of model fit compared to previously estimated
CAPM model with adjusted 𝑅 2 = 0.8691. Also, additional factors SMB and
HML are highly statistically significant with p-values close to zero.
OUTPUT for Question 10:
lm(r_glcgx~rm)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.007049
0.002898 -2.432
0.0181 *
rm
1.167201
0.058906 19.815
<2e-16 ***
--Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.02243 on 58 degrees of freedom
Multiple R-squared: 0.8713, Adjusted R-squared: 0.8691
F-statistic: 392.6 on 1 and 58 DF, p-value: < 2.2e-16
lm(r_glcgx~rm+hml+smb+mom)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.0009467 0.0024567
0.385
0.701
rm
1.0075831 0.0498170 20.226 < 2e-16 ***
hml
-0.4832266 0.0672459 -7.186 1.87e-09 ***
smb
-0.2584757 0.0522464 -4.947 7.47e-06 ***
mom
-0.0018580 0.0313426 -0.059
0.953
--Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.01643 on 55 degrees of freedom
Multiple R-squared: 0.9345, Adjusted R-squared: 0.9297
F-statistic: 196.2 on 4 and 55 DF, p-value: < 2.2e-16
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