An analyst has the following time series data for the United States (US) over the period January 1959 – December 1998

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

Question 1

An analyst has the following time series data for the United States (US) over the period January 1959 – December 1998:

𝑟𝑡= interest rate.

𝑙𝑚𝑡   = logarithm of money.

𝑙𝑝𝑡 = logarithm of prices.

𝑙𝑜𝑡   = logarithm of output

 

The economist aims to analyse the demand for money in the US using a static long run relationship. Figure 1 depicts the dynamics of the four variables while STATA output are given in Table 1:

 

Figure 1

 

 

Table 1

 

lmt Coef. St.Err. t-value p-value [95% Conf Interval]

 

rt *** 0.001 -26.01 0.000 -0.025 -0.022

lpt 0.961 0.014 *** 0.000 0.934 0.989

lot 0.439 0.025 *** 0.000 0.390 0.488

Constant 0.416 0.056 7.38 0.000 0.305 0.527

URSS 1.754 RRSS 249.548

R-squared 0.993 Number of obs 480.000

F-test *** Durbin Watson 0.122

Akaike crit. (AIC) -1323.557 Bayesian crit. (BIC) -1306.862

 

Notes:

1. URSS and RRSS refer to Unrestricted Sum of Squared Residuals and Restricted Sum of Squared Residuals, respectively.

2. Use 5% significance level for all tests.

3. State the null and alternative hypotheses, the test statistic to compute and its distribution, all formulas, and the criteria for rejecting or fail to reject the null hypothesis for all tests.

 

a). Discuss the main statistical properties of the data. Do they exhibit stationarity? Why?

[5 Marks]

b). Write down a mathematical expression describing the true model of the regression estimated in Table 1. What are the main assumptions imposed on this linear model? Discuss the implications of the invalidity of these assumptions.

[7 Marks]

c). Interpret the coefficient estimates in Table 1. Perform a (two-tailed) test of individual significance of the parameters of all the variables using the critical value of the corresponding distribution and the test p-value. Interpret the test results.

[8 Marks]

d). Explain how you would test for the joint significance of the parameters of all variables. What is your conclusion from this test? Provide an interpretation of the goodness of fit of the model.

[5 Marks]

e). The economist formulates a hypothesis that the effect of logarithm output is half the effect of logarithm prices. Perform an F-test for the economist’s hypothesis specifying the null and the equation of the restricted model given that the sum of the squared residuals (RSS) of the restricted model for that test is 1.760. Interpret the test results.

[10 Marks]

f).

i. Define the concept of unit roots? What are the main implications of presence of unit in the data on the estimated model in Table 1?

ii. Describe the steps involved to perform Augmented Dickey Fuller (ADF) test.

iii. Table 2 reports the ADF test for all variables. Discuss the main conclusions of the reported statistics and indicate which variables contain unit roots and which are do not (if any). Explain your answer and show all your working.

[10 Marks]

 

Table 2

 

 

 

g). Test for the presence of serial correlation in the errors formally using the Durbin Watson statistic. What are your conclusions?

[5 Marks]

 

 

 

Questions continue over page

 

Question 2

(a) Given that 𝜀𝑡~𝑖𝑖𝑑(0, 𝜎2), derive the mean, the variance and the covariances of the following processes:

i.

 

 

 

where |𝜃𝑖| < 1, 𝑖 = 1, 2

 

 

ii.

 

 

where 𝑤0 = 0.

 

𝑦𝑡 = 𝜀𝑡 + 𝜃1𝜀𝑡−1 + 𝜃2𝜀𝑡−2,

 

 

 

 

 

𝑤𝑡 = 𝑤𝑡−1 + 𝜀𝑡.

 

 

 

 

[6 Marks]

 

 

 

 

 

[5 Marks]

 

 

 

(b) Which of the above series in (b-i) and (b-ii) are weakly stationary and why?

[4 Marks]

(b) The following regression was run using quarterly data, amounting to 90 observations:

 

𝑏̂𝑡  = 0.49 − 0.27𝑝𝑡 + 0.22𝑦𝑡

(0.84) (0.27) (0.071)

𝑅̅2  = 0.28,  𝐷𝑊 = 1.20, LM(2)=7.42

where (𝑏𝑡) is the demand for text books, (𝑝𝑡) is the price of a book and (𝑦𝑡) is the total level of income, all variables are in logarithms (standard errors in parentheses). DW is the Durbin- Watson test for the first order of autocorrelation and LM(2) is the Lagrange Multiplier test for second order autocorrelation.

 

(i) Does the above regression suffer from first order autocorrelation?

[4 marks]

 

(ii) Briefly Describe how you would conduct the LM test for autocorrelation. Does this model suffer from second order autocorrelation?

[6 marks]

 

Question 3

(a) The following models were fitted to a cross-sectional dataset of 50 firms:

 

 

Model 1

 

 

 

Model 2

 

 

𝑦̂𝑖  = 2.3 + 0.2𝑥1𝑖 − 6.14𝑥2𝑖 − 0.01𝑥3𝑖 + 1.5𝑥4𝑖

(0.052) (1.243) (0.015) (1.324)

𝑅2 = 0.40, 𝑅𝑆𝑆 = 0.60

 

 

𝑦̂𝑖  = 2.6 + 0.25𝑥1𝑖 − 6.14𝑥2𝑖

(0.42) (2.98)

𝑅2 = 0.30, 𝑅𝑆𝑆 = 0.70

 

where y = rate of return on equity for the firm (ROE), x1 = market share, x2 = measure of firm size, x3 = industrial growth rate, x4= level of world trade in industrial products. Numbers in parentheses are coefficient standard errors.

 

i. Based on models 1 and 2, examine the joint hypothesis that the coefficients on x3

and x4 are both equal to zero. Describe all steps.

[5 Marks]

 

ii. Explain how you would test whether financial firms have on average higher ROE.

[5 Marks]

 

 

(b) Suppose you wish test for the presence of ARCH effects in a stock market index. Explain how you would conduct a test for the presence of the 4th order ARCH effects.

[3 Marks]

(c) Using monthly data for 𝑦𝑡, the FTSE100 index, from January 1992 to December 2005, the following set of results were obtained:

 

𝑦̂𝑡  = 0.75,

(0.21)

𝜎̂2 = 1.76 + 0.07𝑢2 + 0.24𝜎2 + 0.87𝑢2 𝐼𝑡−1

 

𝑡 𝑡−1

 

𝑡−1

 

𝑡−1

 

(0.57) (0.02) (0.12) (0.42)

𝑅2 = 0.46

Values in ( ) are the standard errors of the coefficient estimates and 𝐼𝑡−1 = 1 if 𝑢𝑡−1 < 0, 𝐼𝑡−1 = 0 otherwise. 𝜎𝑡−1 is the conditional deviation.

i. What role does the final term in the above conditional variance equation serve?

 

[2 Marks]

 

ii. What is the interpretation of the estimated value of lagged conditional variance (𝜎2 ),

0.24?

 

[5 Marks]

iii. With reference to conditional variance equation, if 𝜎2 = 0.74, consider that 𝑢̂𝑡−1 =

±0.5. Estimate the value of 𝜎2, for a positive shock (+0.5) and a negative shock (-0.5).

 

[5 Marks]

 

 

[END OF QUESTIONS]

 

 

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