SMM248 Statistics for Finance 2021
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MSc in Finance
MSc in Corporate Finance
SMM248
Statistics for Finance
2021
QUESTION 1 [50 marks]
(a) Consider the regression model that a researcher will estimate with T=26 annual observations:
yt = F1 + F2x2t + F3x3t + F4x4t + et
List the ordinary least squares (OLS) regression estimator assumptions and, one by one, discuss: (i) the implications of violating each of them, (ii) how to detect a violation of each of them, (iii) how to remedy each of them.
(8 marks)
(b) There is one salient event in the sample period and the researcher wants to investigate whether two specific parameters, F3 and F4, in model (1) above are stable throughout the sample period. For this purpose, he conducts the Chow breakpoint test by re-formulating the regression equation introducing the dummy variable Dt that takes the value 0 before the event and 1 after the event. Write down: (i) the re-formulated model, (ii) the marginal effect of x3t and x4t on yt before and after the event, (iii) the null hypothesis and the alternative hypothesis for this test so that a test rejection implies that either F3 or F4 have changed; (iv) the name and expression of the test statistic, and the probability distribution that it follows.
(11 marks)
(c) Inspired by finance theory, a researcher conjectures that the marginal effect of x2t on yt is not constant but instead depends on the level of x4t . How shall she reformulate the model equation in order to be able to conduct a test for this effect? Write down: (i) the re-formulated regression model; (ii) the expression of the marginal effect of x2t on yt according to the reformulated model (iii) the null hypothesis and the alternative hypothesis for this test so that a test rejection implies that the researcher was right in his conjecture; (iv) the name and expression of the test statistic, and its probability distribution.
(11 marks)
(d) Suppose that in model (1) above the dependent variable yt is a binary variable that takes the
problems? Write down the expression of the reformulated model and explain how it circumvents
those problems. What is the name of the reformulated model? Can the OLS estimation method be
used to obtain the parameter values of your reformulated model? Give an example of such
regression model (clearly indicate what the variables yt , x2t , x3t and x4t are in your example).
Suppose that the model parameter estimates are F1 = 0, F2 = 0. 5, F3 = 3, F4 = 0. Calculate
according to the reformulated model what is the expected value of yt conditional on these values
(10 marks)
(e) Suppose that a researcher estimates the regression model (1) by OLS and obtains the following results: (i) the R2 of the model is 78%, (ii) the p-value of the F-statistic for the overall significance of the model is 0.03%; (iii) the p-value of the individual significance t-tests for each of the slopes is 0.23%, 0.64% and 0. 12%, respectively. Is there any contradiction in these results? If yes, what is this contradiction likely to be stemming from?
(10 marks)
QUESTION 2 [25 marks]
A researcher is interested in assessing the risk exposure of a particular managed portfolio of UK stocks. She has collected a sample of monthly excess returns on the portfolio (PORTFOLIO) as well as monthly returns on four UK risk factors covering the same time period: the market factor (MARKET) is the excess return on the FTSE 100 index, the size factor (SIZE) is the return on a portfolio of small UK stocks less the return on a portfolio of large UK stocks (where size is measured using market capitalization), the value factor (VALUE) is the return on a portfolio of high value UK stocks less the return on a portfolio of low value UK stocks (where value is measured using the book-to-market ratio), and the momentum (MOMENTUM) factor is the return on UK stocks that have performed strongly over the last year less the return on a portfolio of UK stocks that have performed poorly. All the returns are expressed in decimals. The researcher estimates the multiple linear regression model
yt = a + F1x1t + F2x2t + F3x3t + F4x4t + ct (2)
where yt = P0RTF0LI0; x1t = MARKET; x2t = SIZE; x3t = M0MENTUM; x4t = VALUE; Using the OLS estimation method she obtains the estimation output shown in EXHIBIT 1.
EXHIBIT 1
Dependent Variable: PORTFOLIO
Method: Least Squares; Sample: 1986M10 2016M12;
363 observations
Variable Coefficient Std. Error t-Statistic Prob. |
||||
C |
0.005379 0.001898 2.834781 0.0048 |
|||
MARKET |
0.669732 |
0.039124 |
17.11797 |
0.0000 |
SIZE |
0.833417 |
0.053539 |
15.56653 |
0.0000 |
MOMENTUM |
0.095220 |
0.050715 |
1.877544 |
0.0613 |
VALUE |
0.104161 |
0.055652 |
1.871643
|
0.0621 |
R-squared |
0.602309 |
Mean dependent var |
|
0.010881 |
Adjusted R-squared |
0.597866 |
S.D. dependent var |
|
0.054137 |
S.E. of regression |
0.034331 |
Akaike info criterion |
|
-3.891877 |
Sum squared resid |
|
Schwarz criterion |
|
-3.838235 |
Log likelihood |
711.3757 |
Hannan-Quinn criter. |
|
-3.870555 |
F-statistic |
135.5493 |
Durbin-Watson stat |
|
0.784676 |
Prob(F-statistic) |
0.000000 |
|
|
|
(a) What is the residual sum of squares of the model? What does it represent?
(5 marks)
(b) With the estimation results provided in the above exhibit, can we say anything about whether is correlation in the residuals? Which specific estimation result gives us information about residual autocorrelation? Can we say anything about the order of the autocorrelation? Can we tell if the correlation is positive or negative?
(5 marks)
(c) In order to find if the model above improves upon the simple CAPM model, the researcher specifies a regression excluding the SIZE, MOMENTUM and VALUE factors. The OLS estimation results are shown in EXHIBIT 2 below. She also estimates the same model augmented with the MOMENTUM factor as shown in EXHIBIT 3 below. Construct a table that provides 3 measures or criteria to compare the models in EXHIBITS 1, 2 and 3. Which model shall the researcher select on the basis of this comparison exercise?
(5 marks)
EXHIBIT 2
Dependent Variable: PORTFOLIO
Method: Least Squares;
Sample: 1986M10 2016M12; Included observations: 363
Variable |
Coefficient |
Std. Error |
-Statistic |
Prob. |
|
||||
C MARKET |
0.007468 0.663586 |
0.002347 0.050026 |
3.181679 13.26481 |
0.0016 0.0000 |
R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic) |
0.327691 0.325828 0.044451 0.713298 616.0802 175.9553 0.000000 |
Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat |
0.010881 0.054137 -3.383362 -3.361905 -3.374833 1.398702 |
EXHIBIT 3
Dependent Variable: PORTFOLIO
Method: Least Squares
Sample: 1980M10 2010M12; Included observations: 363
Variable |
Coefficient |
Std. Error |
-Statistic |
Prob. |
|
||||
C |
0.007722 |
0.002397 3.222133 |
0.0014 |
|
MARKET |
0.659565 |
0.050631 13.02694 |
0.0000 |
|
MOMENTUM |
-0.030513 |
0.056758 -0.537591 |
0.5912 |
|
R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic) |
0.328230 0.324498 0.044495 0.712726 616.2259 87.94887 0.000000 |
Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat |
0.010881 0.054137 -3.378655 -3.346470 -3.365862 1.395260 |
(d) An analyst conjectures that the portfolio has a risk exposure to SIZE that is less than 1. Test this conjecture by carrying out a statistical test in the context of the model reported in EXHIBIT
1. Write down the null and alternative hypotheses of the test so that a rejection represents evidence supportive of the analyst’s conjecture. What is the name of the test statistic and the name of the probability distribution that it follows? Draw a graph showing the sample value of the test statistic, 1% critical value of the test and p-value. What shall the analyst conclude?
(5 marks)
(e) In the context of the model in EXHIBIT 3, which probability distribution does the test statistic for the overall significance of the model follow? What is the name of the test statistic? Write down the null and alternative hypotheses of the test in two different ways (in terms of the model parameters; in terms of the coefficient of determination). What is the 1% critical value? Discuss the outcome.
(5 marks)
QUESTION 3 [25 marks]
The market performance of environmentally certified and green commercial buildings and the rent premium they command is a topical research area in finance. In an empirical study investigating the relationship between office rent and the green characteristics of the building where the office is located in central London, the following results are obtained:
ln(RENT) = 3.981 + 0.073RATING – 0.038VACANCY - 0.013SIZE (3)
(0.08) (0.02) (0.01) (0.005)
where the numbers in parentheses are standard errors, the sum of squared residuals (RSS) is 4.980, and the number of central London buildings sampled for the study is 220.
RENT is the achieved rent in £s per square foot reported in CoStar (a leading commercial real estate information company);
RATING is the rating Costar assigns to buildings for their sustainability and green characteristics; the scale of the rating scheme is between 1 and 5, with 1 representing poor environmental specification and 5 excellent green features.
VACANCY is the vacancy in the building in percentage
SIZE is the net leasable area in thousands of square feet of the entire building.
(a) Comment on the impact of the explanatory variables on rents in terms of signs and statistical
significance. Do the signs make sense intuitively? What is the estimate for the rent premium of green buildings according to this study?
(6 marks)
2022-08-20