# Quantitative methods in finance | Mathematics homework help

Fin-40010 Quantitative Methods in Finance Final Assessment SECTION A [60%] Answer all questions 1. Give the formula for the covariance between two random variables and , if and are a. discrete [2 marks] b. continuous. [2 marks] 2. Let Z be a random variable, which is a linear combination of random variables and such that where and are constants. Derive an expression for a. the mean of [2 marks] b. the variance of . [2 marks] 3. Derive, algebraically, the inflation-adjusted one-period simple asset return formula. [2 marks] 4. Suppose the random variable measures the rate of return on a single investment. What is the significance of the mathematical expectation of this random variable in investment decisions? [2 marks] 5. What information does the 4th moment of a probability distribution of a random variable convey? Why this information is considered important in finance? [3 marks] 6. Explain why the normal distribution is not always appropriate for characterizing asset returns? What solution would you propose to this problem? [5 marks] 7. Decide if you agree or disagree with each of the following statements and give a brief explanation of your decision: (a) Like cross-sectional observations, we can assume that most time series observations are independently distributed. [2 marks] (b) A trending variable cannot be used as the dependent variable in multiple regression analysis. [2 marks] (c) Seasonality is not an issue when using annual time series observations. [2 marks] 8. Under which three Gauss-Markov assumptions is the OLS estimator unbiased in a time series regression? Briefly discuss each one. [4 marks] 9. Suppose you have quarterly data on new housing starts, interest rates and real per capita income. Specify a model for housing starts that accounts for possible trends and seasonality in the variables. How would you interpret the estimated coefficients? [5 marks] 10. You are asked to estimate a simple regression model relating the growth in real per capita consumption (of nondurables and services) to the growth in real per capita disposable income. The results are as follows (using the change in the logarithms in both cases): ( ) ( ) where denotes growth in real per capita consumption, denotes growth in real per capita disposable income and standard errors are given in the parentheses. (a) Interpret the equation and discuss statistical significance. [2 marks] (b) Adding a lag of the growth in real per capita disposable income to the equation from part (a) gives the following results: ( ) ( ) ( ) What do you conclude about adjustment lags in consumption growth? [2 marks] (c) If we add the real interest rate in the equation we get the following results: ( ) ( ) ( ) Does the real interest rate affect consumption growth? [2 marks] 11. The file FISH.dta contains daily price and quantity observations on fish prices at the Fulton Fish Market in Manhattan. Contains data from C:Users…stataFISH.DTA obs: 97 vars: 20 19 Nov 2013 23:15 size: 5,141 ————————————————————————————– storage display value variable name type format label variable label ————————————————————————————– — prca float %9.0g price for Asian buyers prcw float %9.0g price for white buyers qtya int %9.0g quantity sold to Asians qtyw int %9.0g quantity sold to whites mon byte %9.0g =1 if Monday tues byte %9.0g =1 if Tuesday wed byte %9.0g =1 if Wednesday thurs byte %9.0g =1 if Thursday speed2 byte %9.0g min past 2 days wind speeds wave2 float %9.0g avg max last 2 days wave height speed3 byte %9.0g 3 day lagged max windspeed wave3 float %9.0g avg max wave hghts of 3 & 4 day lagged hghts avgprc float %9.0g ((prca*qtya) + (prcw*qtyw))/(qtya+qtyw) totqty int %9.0g qtya + qtyw lavgprc float %9.0g log(avgprc) ltotqty float %9.0g log(totqty) t byte %9.0g time trend lavgp_1 float %9.0g lavgprc[_n-1] gavgprc float %9.0g lavgprc – lavgp_1 gavgp_1 float %9.0g gavgprc[_n-1] ————————————————————————————– Sorted by: (a) Based on the following output, is there evidence that price varies systematically within a week? [2 marks] Source | SS df MS Number of obs = 97 ————-+—————————— F( 5, 91) = 1.70 Model | 1.3432706 5 .268654121 Prob > F = 0.1423 Residual | 14.3698396 91 .157910325 R-squared = 0.0855 ————-+—————————— Adj R-squared = 0.0352 Total | 15.7131102 96 .163678231 Root MSE = .39738 —————————————————————————— lavgprc | Coef. Std. Err. t P>|t| [95% Conf. Interval] ————-+—————————————————————- mon | -.0100699 .1293525 -0.08 0.938 -.2670127 .2468729 tues | -.0088125 .1273075 -0.07 0.945 -.2616932 .2440682 wed | .0376262 .1256956 0.30 0.765 -.2120526 .287305 thurs | .090559 .1256707 0.72 0.473 -.1590703 .3401884 t | -.0039912 .0014444 -2.76 0.007 -.0068603 -.0011221 _cons | -.0729573 .1151907 -0.63 0.528 -.3017694 .1558547 —————————————————————————— (b) Adding variables measuring the height of waves over the past several days, results in the following output. Are these variables individually significant? Describe a mechanism by which stormier seas would increase the price of fish. [2 marks] Source | SS df MS Number of obs = 97 ————-+—————————— F( 7, 89) = 5.70 Model | 4.86305046 7 .694721494 Prob > F = 0.0000 Residual | 10.8500597 89 .121910784 R-squared = 0.3095 ————-+—————————— Adj R-squared = 0.2552 Total | 15.7131102 96 .163678231 Root MSE = .34916 —————————————————————————— lavgprc | Coef. Std. Err. t P>|t| [95% Conf. Interval] ————-+—————————————————————- mon | -.018158 .1140693 -0.16 0.874 -.2448113 .2084954 tues | -.0085331 .1121208 -0.08 0.940 -.2313147 .2142485 wed | .0500406 .1117406 0.45 0.655 -.1719856 .2720669 thurs | .1225463 .1109771 1.10 0.272 -.0979629 .3430554 t | -.0011575 .0013908 -0.83 0.408 -.003921 .0016061 wave2 | .0908906 .0217536 4.18 0.000 .0476667 .1341145 wave3 | .0473748 .0208107 2.28 0.025 .0060243 .0887252 _cons | -.9202542 .1898134 -4.85 0.000 -1.297409 -.5430991 —————————————————————————— (c) What happened to the time trend when variables “wave2” and “wave3” were added? How do you explain it? [2 marks] (d) Explain why all explanatory variables in the regression are safely assumed to be strictly exogenous. [2 marks] (e) How would you test the errors of the last regression model for AR(1) serial correlation? Describe the steps and test(s) involved. [4 marks] (f) Based on the following output, is there evidence of serial correlation of order one in the OLS estimates? [2 marks] Source | SS df MS Number of obs = 96 ————-+—————————— F( 1, 94) = 58.13 Model | 4.03955263 1 4.03955263 Prob > F = 0.0000 Residual | 6.53227686 94 .069492307 R-squared = 0.3821 ————-+—————————— Adj R-squared = 0.3755 Total | 10.5718295 95 .111282416 Root MSE = .26361 —————————————————————————— res | Coef. Std. Err. t P>|t| [95% Conf. Interval] ————-+—————————————————————- lres | .6181021 .0810703 7.62 0.000 .457135 .7790691 _cons | .0088355 .0269086 0.33 0.743 -.0445922 .0622632 —————————————————————————— (g) How would you correct for serial correlation in this case? [5 marks] SECTION B (40%) Answer all questions 12. What are the main assumptions underlying the basic OLS regression model. Outline the importance of each assumption for OLS estimation and discuss how would you empirically test each assumption? [20 marks] 13. A partial adjustment model is: ( ) Where is the desired or optimal level of , and is the actual (observed) level. The second equation describes how the actual adjusts depending on the relationship between the desired in time and the actual in time . (a) How would you interpret parameters and ( )? [5 marks] (b) Show that we can write i.e. find in terms of and , and in terms of and . [5 marks] (c) If ( | ) ( | ) and all series are weakly dependent, how would you estimate the ? [5 marks] (d) If ̂ and ̂ , what are the estimates of and ? [5 marks] END

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