Midterm Exam

Economics 704 (Econometrics)

Formulas:

tˆ = βˆ − β ∗

se (  )βˆ

The F-test statistic is given by

 = RRSS − URSS T − k

URSS            k

where URSS = RSS from unrestricted regression, RRSS = RSS from re- stricted regression, m = number of restrictions, T = number of observations, k = number of regressors in unrestricted regression

1.- Assume that the relationship between a company’s stock price (y) and dividends paid per share (x) is linear.  If the slope of the equation is 0.50 and the intercept is 30, what would be the expected stock price if the dividend paid was 3?

It would be

E (y | x) = 30 + 0.5 × 3 = 31.5

2.- What does the central limit theorem state?

That the sampling distribution of the mean of any random sample of ob- servations will tend towards the normal distribution with mean equal to the population mean as the sample size tends to infinity

3.- What do we call data that have been collected over a period of time on one or more variables?

Time-series data

4.- In a time-series regression of the excess return of a mutual fund on a constant and the excess return on a market index, when would we say that the the fund manager is considered to have‘beaten the market’in a statistical sense?

The estimate for the intercept α should be positive and statistically signifi- cant.

5.- Explain why the following statement is either true or false:  A‘95% confidence interval’for a regression parameter means that, in repeated samples, we would derive the same estimate for the coefficient 95% of the time.

The statement is false, what the confidence interval means is that in repeated

samples 95% of the time the interval formed as βˆ±t−table95% se (  )βˆ will contain the true β . Another way of saying it is that we (from our perspective) are 95% certain that the true β is contained in the interval.

6.- Two researchers have identical models, data, coefficients and standard error estimates.  They test the same hypothesis using a two-sided alternative, but researcher 1 uses a 5% size of test while researcher 2 uses a 10% test. Which one will be more likely to make a type-1 error. Explain your answer.

Since the size of a test is the probability that we reject the null hypothesis when it is in fact true, and this is the definition of the type 1 error, the second researcher is twice as likely to make this mistake.

7.- Consider the following regression estimated using 84 observations:

yt  = 1 + 2X2t + 3X3t + 4X4t + ut

Suppose that a researcher wishes to test the null hypothesis:  β2  = 1 and β3 + β4  = 1. The table value of the F-distribution that we would compare the result of testing this hypothesis with at the 10% level would be approximately

We would have an F (2, 80) so looking at the tables we would have a number between 2.35 and 2.39, so approx: 2.37.

8.- What is the long-run solution to the following dynamic econometric model?

∆Yt  = β 1 + β2 ∆X2t + β3 X3t + ut

There is no long-run solution to these model as it only tells us something about the chnages in Yt  so we cannot determine the level of Yt .  Mechanically you can see that when you set the changes in the variables to zero we are left with nothing.

9.- Imagine you have the following regression model

yt  = β 1 + β2 xt + ut

but the errors are autocorrelated. In particular you know that

ut  = ρ0 + ρ 1 ut −1 + εt

where E (εt  | xt ,xt −1,yt −1) = 0.

i) Show that you can rewrite the model as

yt  = α 1 + α2 xt + α3 xt −1 + α4 yt −1 + ωt

To see why first notice that we also have

yt −1  = β 1 + β2 xt −1 + ut −1 .

and that we can write

yt  = β 1 + β2 xt + ρ0 + ρ 1 ut −1 + εt

Now solve for ut −1  in teh second equation and and replace above

yt  = β 1 + β2 xt + ρ0 + ρ 1 [yt −1 − β1 − β2 xt −1] + εt

= [β1 + ρ0 − ρ1 β 1] + β2 xt − ρ1 β2 xt −1 + ρ 1 yt −1 + εt

which the results we wanted