Econometrics

In the regression model y=β0 +β1x+β2d+β3(x×d)+u, where x is a continuous variable and d is a dummy variable, β2

indicates the difference in the intercept when d = 1 compared to the base group.

16. In the regression model y=β0 +β1x+β2d+β3(x×d)+u, where x is a continuous variable and d is a dummy variable, β3

indicates the difference in the slope parameter when d = 1 compared to the base group.

The population correlation coefficient between two variables X and Y

is a measure of linear association.

The expected value of a discrete random variable:

is computed as a weighted average of the possible outcome of that random variable, where the weights are the probabilities of that outcome.

A type II error

is the error you make when not rejecting the null hypothesis when it is false.

The size of the test

is the probability of committing a type I error.

The OLS estimator is derived by

minimizing the sum of squared residuals.

The power of the test is

one minus the probability of committing a type II error.

A manufacturer claims that his tires last at least 40,000 miles. A test on 25 tires reveals that the mean life of a tire is 39,750 miles, with a standard deviation of 387 miles. Compute the actual value of the t statistic.

t = −3.23.

The cumulative probability distribution shows the probability:

that a random variable is less than or equal to a particular value.

You have to worry about perfect multicollinearity in the multiple regression model because

the OLS estimator cannot be computed in this situation.

In the simple linear regression model y = β0+β1x+u E (u|x) = 0 the regression slope

indicates by how many units the conditional mean of y increases, given a one unit increase in x.

Let R2 be the R-squared of a regression (that includes a constant), SST be the total sum of squares of the dependent variable, SSR be the residual sum of squares and 2 df be the degrees of freedom. The estimator of the error variance, σ = SSR/df, can be re-written as:

(1−R²)SST /df

For n = 25, sample mean =645, and sample standard deviation s = 55, construct a 99% confidence interval for the population mean.

(614.23, 675.77)

For n = 121, sample mean=96, and a known population standard deviation σX = 14, construct a 95% confidence interval for the population mean.

(93.51, 98.49).

Consider the following estimated model (by OLS), where return is the total return of holding a firm's stock during one year, dkr is the firm's debt to capital ratio, eps denotes earnings per share, netinc denotes net income and salary denotes total compensation, in millions of dollars, for the CEO (estimated standard errors of the parameters in parentheses below the estimates). The model was estimated using data on n = 142 firms. return = −12.3 + 0.32 dkr + 0.043eps − 0.005 netinc + 0.0035salary, (6.89) (0.150) (0.078) (0.0047) (0.0022) n = 142, R2 = 0.0395 Which of the following is the 99% confidence interval for the coefficient on dkr?

(−0.0664,0.7064)

A spark plug manufacturer believes that his plug lasts an average of 30,000 miles, with a standard deviation of 2,500 miles. What is the probability that a given spark plug of this type will last 37,500 miles before replacement? Assume a normal distribution.

0.0013

A box has 20 screws, three of which are known to be defective. What is the probability that the first two screws taken out of the box are both defective?

0.0158

Consider the following estimated model (by OLS), where return is the total return of holding a firm's stock during one year, dkr is the firm's debt to capital ratio, eps denotes earnings per share, netinc denotes net income and salary denotes total compensation, in millions of dollars, for the CEO (estimated standard errors of the parameters in parentheses below the estimates). The model was estimated using data on n = 142 firms. return = −12.3 + 0.32 dkr + 0.043eps − 0.005 netinc + 0.0035salary, (6.89) (0.150) (0.078) (0.0047) (0.0022) n = 142, R2 = 0.0395 What is the correlation between the fitted values and the dependent variable?

0.1987

Let Z be a standard normal random variable. Find Pr(-0.5 < Z < 0.5).

0.3830

The probability of stock A rising is 0.3; and of stock B rising is 0.4. What is the probability that neither of the stocks rise, assuming that these two stocks are independent?

0.42

Find the probability that a standard normal random variable has a value greater than -1.56.

0.9406

Given the following probability distribution: X P(X) 1 0.2 2 0.3 3 0.3 4 0.2 What is the variance of the random variable X?

1.05

A bag has five pearls in it, out of which one is artificial. If three pearls are taken out at random, what is the probability that the artificial pearl is one of them?

3/5

Let X be a normally distributed random variable with mean 100 and standard deviation 20. Find two values, a and b, symmetric about the mean, such that the probability of the random variable being between them is 0.99.

48.5, 151.5

Consider the following estimated model (standard errors in parentheses) wage = 235.3923 + 60.87774educ − 2.216635hours, (104.1423) (5.716796) (1.738286) n = 935, R2 = 0.108555, where wage is the wage in euros, educ is the level of education measured in years and hours is the average weekly hours of work. What is the F-statistic for the overall significance of the model.

56.747

Denote the R2 of the unrestricted model by R2 and the R2 of the restricted model by R2 . Let R2 and R2 be 0.4366 and 0.4149 respectively. The difference R UR R between the unrestricted and the restricted model is that you have imposed two restrictions. The unrestricted model has one intercept and 3 regressors. There are 420 observations. The F-statistic in this case is

8.01

Of the following assumptions, which one(s) is (are) necessary to guarantee unbi- asedness of the OLS estimator in a multiple linear regression context? a) Linearity of the model in the parameters. b) Zero conditional mean of the error term. c) Absence of perfect multicollinearity. d) Homoskedasticity of the error term. e) Random sampling.

All of the above except d).

In a random sample:

All the individuals or units from the population have the same probability of being chosen.

In a multiple linear regression where the Gauss-Markov assumptions hold, why can you interpret each coefficient as a ceteris paribus effect?

Because the Ordinary Least Squares (OLS) estimator of the coefficient on variable xj is based on the covariance between the dependent variable and the variable xj after the effects of other regressors has been removed.

X and Y are two random variables. Which of the following statements holds true regardless of whether X and Y are independently distributed?

E(Y ) = E[E(Y |X)]

An estimator θ of the population value θ is unbiased if

E(θ₋) = θ.

Let Y be a random variable with mean μY . Then var(Y ) equals:

E[(Y − μY )2].

Consider the model: log(price) = β0 + β1score + β2breeder + u, where price is the price of an adult horse, score is the grade given by a jury (higher score means higher quality of the horse) and breeder is the reputation of the horse breeder. The estimated model is: log(price) = 5.84 + 0.21score + 0.13breeder What is the interpretation of the estimated coefficient on score?

Each additional grade point increases the horse's price by 21%, on average, ceteris paribus.

The conditional expectation of Y given X, E[Y |X = x], is calculated as follows:

Eki=1 yi Pr(Y = yi|X = x).

Denote SSRr the sum of squared residuals of a restricted model and SSRur the sum of squared residuals of an unrestricted model, n the sample size and k the number of regressors of the unrestricted model. All the following are correct formulae for the F-statistic for testing q restrictions with the exception of

F = ((SSRur−SSRr)/q)/(SSRr/(n-k-1))

In the regression model y=β0 +β1x+β2d+β3(x×d)+u, where x is a continuous variable and d is a dummy variable, to test that the intercept and slope parameters for d = 0 and d = 1 are identical, you must use the

F-statistic for the joint hypothesis that β2 = 0, β3 = 0.

Consider the following estimated model (by OLS), where return is the total return of holding a firm's stock during one year, dkr is the firm's debt to capital ratio, eps denotes earnings per share, netinc denotes net income and salary denotes total compensation, in millions of dollars, for the CEO (estimated standard errors of the 3 parameters in parentheses below the estimates). The model was estimated using data on n = 142 firms. return = −12.3 + 0.32 dkr + 0.043eps − 0.005 netinc + 0.0035salary, (6.89) (0.150) (0.078) (0.0047) (0.0022) n = 142, R2 = 0.0395 What can you say about the estimated coefficients of the variable salary? (con- sider a two-sided alternative for testing significance of the parameters)

For each additional million dollars in the wage of the CEO, return is predicted to increase by 0.0035, on average, ceteris paribus. But it is not statistically significant at the 5% level of significance.

Consider the following estimated model (standard errors in parentheses) wage = 235.3923 + 60.87774educ − 2.216635hours, (104.1423) (5.716796) (1.738286) n = 935, R2 = 0.108555, where wage is the wage in euros, educ is the level of education measured in years and hours is the average weekly hours of work. What can you say about the estimated coefficient of the variable educ? (consider a two-sided alternative for testing significance of the parameters)

For each additional year of education, wage is predicted to increase by 60.88 euros, on average, ceteris paribus. But it is statistically significant at a 5% level of significance.

In the estimated model log(q)=502.57−0.9log(p)+0.6log(ps)+0.3log(y),where p is the price and q is the demanded quantity of a certain good, ps is the price of a substitute good and y is disposable income, what is the meaning of the coefficient on p? (Assume that the Gauss-Markov assumptions hold in the theoretical model)

If the price increases by 1%, the demanded quantity will be 0.9% lower on average, ceteris paribus.

In the estimated model log(q)=502.57−0.9log(p)+0.6log(ps)+0.3log(y),where p is the price and q is the demanded quantity of a certain good, ps is the price of a substitute good and y is disposable income, what is the meaning of the coefficient on ps?

It is the cross-price elasticity of demand in relation to the substitute good and it bears the expected sign.

In the model Grade = β0 + β1study + β2leisure + β3sleep + β4work + u, where each regressor is the amount of time (hours), per week, a student spends in each one of the named activities and where the time allocation for each activity is explaining Grades (where Grade if the final grade of Introduction to Econo- metrics), what assumption is necessarily violated if the weekly endowment of time (168 hours) is entirely spent either studying, or sleeping, or working, or in leisure activities?

No perfect multicollinearity.

In the simple regression model an unbiased estimator for V ar(u) = σ2, the variance of the population regression errors, is:

SSR/(n−2).

Which of the following statements is correct?

SST = SSE + SSR

Under the assumption of the Gauss-Markov Theorem, in the simple linear regres- sion model, the OLS estimator is BLUE. This means what?

The OLS estimator is the estimator that has the smallest variance in the class of linear unbiased estimators of the parameters.

Consider the model: log(price) = β0 + β1score + β2breeder + u, where price is the price of an adult horse, score is the grade given by a jury (higher score means higher quality of the horse) and breeder is the reputation of the horse breeder. Because reputation of the breeder is difficult to measure we decided to estimate the model omitting the variable breeder. What bias can you expect in the score coefficient, assuming breeder reputation is positively correlated with score and β2 > 0?

The estimated coefficient of score will be biased upwards.

Consider a regression model where the R2 is equal to 0.2359 with n = 46 and k = 5. You are testing the overall significance of the regression using the F -Test. What is the p-value of the test?

The p-value is approximately 5% (but a little bit smaller than that).

The error term is homoskedastic if

V ar(u|x) is constant

Consider a regression model where the R2 is equal to 0.257 with n = 33 and k = 3. You are testing the overall significance of the regression using the F-Test. Which of the statements below is correct?

We can reject the null hypothesis at the 5% level of significance but not at the 1% level of significance.

Take an observed (that is, estimated) 95% confidence interval for a parameter of a multiple linear regression. Then:

We cannot assign a probability to the event that the true parameter value lies inside that interval.

What does it mean when you calculate a 95% confidence interval?

Were the procedure you used to construct the confidence interval to be re- peated on multiple samples, the calculated confidence interval (which would differ for each sample) would encompass the true parameter 95% of the time.

An estimate is:

a nonrandom number.

An estimator is:

a random variable.

Var(aX + bY ) is

a2σX2 +2abσXY +b2σY2

We would like to predict sales from the amount of money insurance companies spent on advertising. Which would be the independent variable?

advertising.

The overall regression F-statistic tests the null hypothesis that

all slope coefficients are zero

In the Chow test the null hypothesis is:

all the coefficients in a regression model are the same in two separate populations.

The correlation between X and Y:

can be calculated by dividing the covariances between X and Yby the product of the two standard deviations.

Suppose that n = 100, and that we want to test at 1% level whether the population mean is equal to 20 versus the alternative that it is not equal to 20. The sample mean is found to be 18 and the sample standard deviation is 10. Your conclusion is:

do not reject the null hypothesis.

Let σX2 be the population variance and X be the sample mean. V ar(X) is:

equal to σX2 divided by the sample size.

When there are ∞ degrees of freedom, the t∞ distribution:

equals the standard normal distribution.

Omitted variable bias

exists if the omitted variable is correlated with the included regressor and is a determinant of the dependent variable.

In the simple regression model y = β0 + β1x + u, to obtain the slope estimator using the least squares principle, you divide the

sample covariance of x and y by the sample variance of x.

When testing joint hypotheses, you should

se the F-statistics and conclude that at least one of the restrictions does not hold if the statistic exceeds the critical value.

In testing multiple exclusion restrictions in the multiple regression model under the classical assumptions, we are more likely to reject the null that some coefficients are zero if:

the R-squared of the unrestricted model is large relative to the R-squared of the restricted model.

The Student t distribution is:

the distribution of the ratio of a standard normal random variable, divided by the square root of an independently distributed chi−squared random variable with m degrees of freedom divided by m.

A type I error is

the error you make when rejecting the null hypothesis when it is true.

The proof that OLS is BLUE (Gauss-Markov Theorem) requires all of the following assumptions with the exception of:

the errors are normally distributed.

The regression R2 is a measure of

the goodness of fit of your regression line.

15. Which of the following is not correct in a regression model containing an interaction term between two independent variables, x1 and x2:

the interaction term coefficient is the effect of a unit increase in √x₁x₂.

Take an observed (that is, estimated) 95% confidence interval for a parameter of a multiple linear regression. If you increase the confidence level to 99% then, necessarily:

the length of the confidence interval increases.

In testing multiple exclusion restrictions in the multiple regression model under the Classical Linear Model assumptions, we are more likely to reject the null that some coefficients are zero if:

the residuals sum of squares of the restricted model is large relative to that of the unrestricted model.

The OLS slope estimator, β1, has a smaller standard error, other things equal, if

there is more variation in the explanatory variable, x

Let y be the fitted values. The OLS residuals, u , are defined as follows:

y −y₋

In the simple regression model y = β0 + β1x + u, the simple average of the OLS residuals is

zero

Assume that Y is normally distributed N(μ,σ2). To find Pr(c1 ≤ Y ≤ c2), where c1 <c2,anddi =ci−μ,youneedtocalculatePr(d1 ≤Z≤d2)=

Φ(d2 ) − Φ(d1 )

Suppose we have the linear regression model y = β0 + β1x1 + β2x2 + u, and we would like to test the hypothesis H0 : β1 = β2. Denote β1 and β2 the OLS estimators of β1 and β2. Which of the following statistics can be used to test H0?

β⁻₁-β⁻₂/(Var(β⁻₁)- 2Cov(β⁻₁, β⁻₂)+ Var(β⁻₂))

In the multiple regression model the standard error of regression is given by

√((1/(n-k-1)∑(n, 1=i)⁻u²))