TEST 1

Short Questions

1. Assume you want to estimate the causal effect of $X$ on $Y$. The notion of ceteris paribus plays an important role in causal analysis. It means:

• Absence of other relevant factors.
• Averages of all other relevant factors are zero.
• All other relevant factors being equal.
• All other relevant factors are correlated with $X$.

Concept: Ceteris Paribus

A ceteris paribus comparison changes $X$ while keeping the other determinants of $Y$ at the same values in the situations being compared. Here, “being equal” means “unchanged across the comparison”; it does not mean that the other factors are equal to one another.

If

where $Z$ contains the other relevant determinants of $Y$, the ceteris paribus effect of $X$ is

with $Z$ fixed.

Solution:

The correct option is all other relevant factors being equal.

• The other factors do not need to be absent.
• Their averages do not need to be zero.
• Correlation between them and $X$ is a separate issue that can create confounding.

Answer: All other relevant factors being equal, meaning held constant across the comparison.

2. Suppose that you are asked to evaluate whether more job training makes workers more productive. You observe hours of job training per worker, $training$, and the number of nondefective items produced per worker-hour, $output$. If you find a positive correlation between output and training, have you convincingly established that training makes workers more productive? Explain.

Concept: Association versus Causation

A positive correlation shows that higher training and higher output occur together in the sample. It identifies a causal effect only if training is unrelated to the other determinants of output contained in the error term.

Consider

A sufficient condition for interpreting $\beta_1$ causally is

or the corresponding conditional version when control variables are included.

Solution:

The positive correlation may reflect factors such as worker ability, prior experience, firm quality, machinery, or management practices. These factors may affect both training and output. Reverse causality is also possible: workers who are already more productive may be selected for more training.

Therefore,

shows a positive association, but it does not separate the effect of training from these alternative explanations.

Answer: No. A causal conclusion requires a design or assumption that makes training exogenous, such as random assignment or a defensible zero conditional mean assumption.

3. The OLS regression line relating monthly wage to years of education is

Answer the following:

1. How do you interpret the intercept?
2. What do you conclude by comparing predicted wages at education levels equal to $13$ and $18$?
3. How do you interpret the slope coefficient?
4. Could you conclude that more education causes higher wages?
5. Could you conclude that changes in education are statistically significant in affecting wages?

Concept: Level-Level Regression

In

$\hat\beta_0$ is the predicted value of $Y$ at $X=0$, and $\hat\beta_1$ is the predicted change in $Y$ from a one-unit increase in $X$.

Step-by-step Solution:

1. Intercept

At $educ=0$,

The fitted model predicts a monthly wage of $600$ for zero years of education. This interpretation may have limited practical value if zero education is outside the observed sample range.

2. Predicted wages at $13$ and $18$ years

The predicted difference is

3. Slope

One additional year of education is associated with an increase of $80$ units in predicted monthly wage.

4. Causality

The regression alone does not establish causality. Education may be correlated with omitted wage determinants such as ability, family background, occupation, location, or experience. A causal interpretation requires an assumption such as

or another identification strategy.

5. Statistical significance

To test

we need

No standard error or equivalent inference information is given, so statistical significance cannot be determined.

Answers:

1. Predicted wage at zero years of education: $600$.
2. Predicted wages: $1640$ and $2040$; difference: $400$.
3. One more year of education is associated with $80$ additional wage units.
4. No causal conclusion follows from the fitted line alone.
5. Statistical significance cannot be assessed without a standard error or test statistic.

Exercises

4. The OLS regression line using the logarithm of wage as the dependent variable is

where $female$ is a dummy variable identifying female workers, $educ$ is years of education, and $exper$ is experience. What does the coefficient on $female$ imply?**

Concept: Dummy Variable with a Logarithmic Dependent Variable

In a model with $\log(Y)$ as the dependent variable and a dummy $D$, the coefficient on $D$ is the fitted log difference between $D=1$ and the reference group $D=0$, holding the other regressors fixed.

The approximate percentage difference is $100\hat\beta_D\%$. The exact fitted percentage difference is

Step-by-step Solution:

For male workers, $female=0$:

For female workers, $female=1$:

Thus, holding education and experience fixed,

Approximation:

Exact transformation:

Answer: Holding education and experience fixed, the fitted wage for female workers is approximately $25.7\%$ lower than for male workers using the exact transformation. The common linear approximation gives $29.7\%$ lower. This is a conditional association unless the regression has a causal interpretation.

5. The OLS regression line using wage as the dependent variable is

where $female$ is a dummy variable identifying female workers and $educ$ is years of education.

1. According to the regression line, what is your prediction regarding the effect of five more years of education?
2. Assume that the standard error of the coefficient on the interaction term is $13$. What do you conclude about its statistical significance?

Concept: Dummy-Continuous Interaction

For

$\beta_1$ is the slope for $D=0$, while $\beta_1+\beta_3$ is the slope for $D=1$. The interaction coefficient $\beta_3$ is the difference between the two slopes.

Step-by-step Solution:

1. Five-year effect

For male workers, $female=0$:

The education slope is $80$, so five more years change predicted wage by

For female workers, $female=1$:

The education slope is $80-30=50$, so five more years change predicted wage by

2. Test of the interaction

Test

The test statistic is

Answer: Five more years of education increase predicted wage by $400$ for male workers and $250$ for female workers. The interaction has $t\approx-2.31$; it is significant at the usual two-sided $5\%$ level in a moderate or large sample, but the exact finite-sample decision requires the degrees of freedom.

TEST 2

Short Questions

1. The OLS regression line using the logarithm of wage as the dependent variable is

where $male$ is a dummy variable identifying male workers, $educ$ is years of education, and $exper$ is experience. What does the coefficient on $male$ imply?**

Concept: Reference Group

The group coded zero is the reference group. Since $male=1$ identifies male workers, female workers form the reference group.

Step-by-step Solution:

For female workers, $male=0$:

For male workers, $male=1$:

The fitted log-wage difference is

The exact fitted percentage difference is

Answer: Holding education and experience fixed, the fitted wage for male workers is approximately $25.7\%$ lower than for female workers. The linear approximation is $29.7\%$ lower.

Exercises

2. The OLS regression line using wage as the dependent variable is

where $male$ is a dummy variable identifying male workers and $educ$ is years of education.

1. According to the regression line, what is your prediction regarding the effect of five more years of education?
2. Assume that the standard error of the coefficient on the interaction term is $13$. What do you conclude about its statistical significance?

Concept: Group-Specific Slopes

The coefficient on $educ$ is the slope for the reference group, $male=0$. The interaction coefficient gives the change in that slope when $male=1$.

Step-by-step Solution:

For female workers, $male=0$:

Five more years of education change predicted wage by

For male workers, $male=1$:

Five more years of education change predicted wage by

For the interaction coefficient,

The exact decision depends on the residual degrees of freedom. Using the large-sample two-sided $5\%$ critical value, the coefficient is significant; using the $1\%$ critical value, it is not.

Answer: Five additional years of education increase predicted wage by $250$ for female workers and $400$ for male workers. The interaction has $t\approx2.31$ and is significant at the usual two-sided $5\%$ level in a moderate or large sample.

TEST 3

Short Questions

1. Which of the following can cause the usual OLS $t$-statistics to be invalid—that is, not to have $t$-distributions under $H_0$?

• Heteroskedasticity.
• A sample correlation coefficient close to zero between two independent variables of the model.
• Omitting a relevant explanatory variable.

Concept: Conditions for the Conventional OLS t-Test

The conventional OLS $t$-statistic uses the homoskedastic variance estimator. Its exact finite-sample $t$ distribution also uses conditional normality. A misspecified error variance or a biased coefficient can invalidate the test.

Solution:

• Heteroskedasticity: Yes. If $\operatorname{Var}(u_i\mid X_i)$ is not constant, the conventional homoskedastic standard error is generally incorrect. Robust standard errors are needed.
• Regressor correlation close to zero: No. This does not violate an OLS assumption.
• Omitting a relevant explanatory variable: It invalidates inference for the intended coefficient when the omitted variable affects $Y$ and is correlated with an included regressor. In that case, zero conditional mean fails.

Answer: Heteroskedasticity, and omitting a relevant explanatory variable when it is correlated with an included regressor. A correlation close to zero between regressors does not invalidate the test.

2. Discuss briefly the classical linear model assumptions for cross-sectional regression.

Concept: Classical Linear Model

For

the assumptions can be separated by the result they support.

Solution:

1. Linearity in parameters

   \[Y_i=\beta_0+\beta_1X_{1i}+\cdots+\beta_kX_{ki}+u_i.\]

   Regressors may include logs, squares, and interactions; the model must be linear in the coefficients.

2. Random sampling

   The observations $(Y_i,X_{1i},\ldots,X_{ki})$, $i=1,\ldots,n$, are randomly sampled from the population.

3. No perfect multicollinearity

   No regressor is an exact linear combination of the others. In matrix form,

   \[\operatorname{rank}(\mathbf X)=k+1.\]

4. Zero conditional mean

   \[E(u_i\mid X_{1i},\ldots,X_{ki})=0.\]

   This is the main condition for conditional unbiasedness of OLS.

5. Homoskedasticity

   \[\operatorname{Var}(u_i\mid X_{1i},\ldots,X_{ki})=\sigma^2.\]

   Together with the first four assumptions, this gives the Gauss-Markov efficiency result and validates the conventional homoskedastic variance formula.

6. Conditional normality

   \[u_i\mid\mathbf X\sim N(0,\sigma^2).\]

   This gives exact finite-sample $t$ and $F$ distributions. It is not required for unbiasedness.

Summary: Assumptions 1–4 give conditional unbiasedness; adding homoskedasticity gives the Gauss-Markov result; adding normality gives exact finite-sample inference.

Exercises

3. Consider the cross-sectional model relating the annual number of crimes on college campuses, $crime$, to student enrollment, $enroll$:

1. How do you interpret the estimated slope coefficient?
2. Assume that the standard error of the estimated slope is $0.11$. Construct a testing strategy for the null hypothesis that the slope coefficient equals $1$.

Concept: Log-Log Model and Test Against a Nonzero Value

In

$\beta_1$ is the elasticity of $Y$ with respect to $X$.

Step-by-step Solution:

1. Slope interpretation

The estimate $1.27$ means that a $1\%$ increase in enrollment is associated with approximately a $1.27\%$ increase in the number of crimes. The statement is an association unless the model supports a causal interpretation.

2. Hypothesis test

Use a two-sided test:

The test statistic is

With an intercept and one slope, the exact reference distribution under the classical normal assumptions is $t_{n-2}$. The sample size is not given, so the exact critical value is unknown.

Using the large-sample two-sided $5\%$ critical value $1.96$,

so $H_0$ is rejected. Using the large-sample $1\%$ critical value $2.576$, it is not rejected.

Answer: The estimated elasticity is $1.27$. The test statistic for $H_0:\beta_1=1$ is $2.45$. The null is rejected at the usual two-sided $5\%$ level for a moderate or large sample; the exact finite-sample decision requires $n$.

4. Consider the time-series model

Assume that the model is estimated by OLS and that $\hat\beta$ estimates $\beta$. Discuss useful assumptions implying the unbiasedness of $\hat\beta$ and prove the unbiasedness.**

Concept: Strict Exogeneity and Unbiasedness

Because the time-series OLS estimator depends on the full sequence $X_1,\ldots,X_T$, a sufficient condition for finite-sample unbiasedness is

Assumptions:

1. $Y_t=\alpha+\beta X_t+\varepsilon_t$ is correctly specified.
2. $\sum_{t=1}^{T}(X_t-\bar X)^2>0$.
3. Strict exogeneity holds.
4. The required expectations are finite.

Homoskedasticity, normality, and absence of serial correlation are not needed for this unbiasedness result.

Proof:

The OLS slope is

Substituting $Y_t=\alpha+\beta X_t+\varepsilon_t$ and using

gives

Let $\mathbf X=(X_1,\ldots,X_T)$. Conditional on $\mathbf X$, the weights are fixed, so

By iterated expectations,

Answer: Under strict exogeneity and variation in $X$, $\hat\beta$ is conditionally and unconditionally unbiased.

TEST 4

Short Questions

1. Which of the following can cause the usual OLS $t$-statistics to be invalid—that is, not to have $t$-distributions under $H_0$?

• Serial correlation.
• A sample correlation coefficient of $0.95$ between two independent variables of the model.
• Omitting a relevant explanatory variable.

Concept: Error Dependence, Multicollinearity, and Omitted Variables

Serial correlation affects the conventional variance formula. High but imperfect multicollinearity affects precision. Omitted-variable bias arises when an omitted determinant is correlated with an included regressor.

Solution:

• Serial correlation: Yes. If $\operatorname{Cov}(u_t,u_s\mid\mathbf X)\neq0$ for some $t\neq s$, conventional OLS standard errors are generally incorrect. A serial-correlation-robust method is required.
• Correlation of $0.95$ between regressors: No, not by itself. The correlation is high but not perfect. It can produce large standard errors and unstable coefficient estimates, but it does not invalidate the $t$ distribution when the remaining assumptions hold.
• Omitting a relevant explanatory variable: Yes when the omitted variable is correlated with an included regressor. Then zero conditional mean fails and the coefficient does not estimate the intended parameter without bias.

Answer: Serial correlation, and omitting a relevant explanatory variable when it is correlated with an included regressor. A correlation of $0.95$ causes multicollinearity but is not, by itself, a failure of the distributional assumptions.

2. Discuss the notion of strict exogeneity.

Concept: Strict Exogeneity in Time Series

For

strict exogeneity requires

Discussion:

This condition says that the error in period $t$ has conditional mean zero given past, present, and future values of the regressor. It rules out feedback from a current error to future values of $X$.

For example, if an unexpected sales shock at time $t$ changes advertising at time $t+1$, then $u_t$ is related to $X_{t+1}$ and strict exogeneity fails.

Strict exogeneity is stronger than contemporaneous exogeneity,

which only restricts the within-period relationship. It is also stronger than predeterminedness, which allows future regressors to respond to current shocks.

Strict exogeneity does not require $X_t$ to be serially independent. It also does not by itself require homoskedastic or serially uncorrelated errors; those conditions concern the variance and conventional inference.

Consequence: Under the remaining linearity and rank conditions,

Exercises

3. Consider the model

Assume that the model is estimated by OLS and that $\hat\beta$ estimates $\beta$. Derive the variance of $\hat\beta$.**

Concept: Partial Variation in a Multiple Regression

The precision of $\hat\beta$ depends on the part of $X$ that cannot be explained by the intercept and $Z$. Under homoskedasticity, more residual variation in $X$ gives a smaller variance.

Assume

and no perfect collinearity.

Step-by-step Solution:

Regress $X_i$ on an intercept and $Z_i$:

The residuals $r_i$ are the part of $X_i$ orthogonal to the intercept and $Z_i$. By the Frisch-Waugh-Lovell result,

Conditional on $X$ and $Z$,

Let

and let $R_X^2$ be the $R^2$ from the auxiliary regression of $X$ on an intercept and $Z$. Since

we obtain

With only one additional regressor, $R_X^2=r_{XZ}^2$, where $r_{XZ}$ is the sample correlation between $X$ and $Z$. Therefore,

Equivalently, with

The variance rises when $\sigma^2$ rises or when $X$ becomes more predictable from $Z$. If $R_X^2=1$, perfect multicollinearity prevents separate estimation of $\beta$.

The estimated variance replaces $\sigma^2$ with

TEST 5

Short Questions

1. Consider the regression equations

and

where $\varepsilon_i$ and $u_i$ are error terms.

Provide interpretations of the OLS estimates $\hat\beta_1$ and $\hat\beta_2$, and derive the equation expressing the relationship between the two estimates.**

Concept: Included and Omitted Regressors

$\hat\beta_1$ is the partial slope of $Y$ on $X$ holding $Z$ fixed. $\hat\beta_2$ is the slope from the simple regression that does not hold $Z$ fixed.

Step-by-step Derivation:

Define

From the fitted multiple regression,

Multiply by $X_i-\bar X$ and sum. The intercept term drops out, and OLS gives

Hence

The simple-regression slope is

Therefore,

Let $\hat\delta$ be the slope from the auxiliary regression

Then

so

Equivalently,

Answer: The difference between the simple- and multiple-regression slopes equals the estimated effect of $Z$ on $Y$, holding $X$ fixed, multiplied by the sample association between $Z$ and $X$.

2. Consider the regression equation

where $Y$ is monthly wage and $F_i$ is a dummy variable identifying a subset of observations, such as female employees.

1. Interpret the OLS estimates $\hat\alpha$ and $\hat\beta$.
2. Now consider \(Y_i=\gamma_1M_i+\gamma_2F_i+\varepsilon_{2i},\) where $M_i=1-F_i$. Interpret $\hat\gamma_1$ and $\hat\gamma_2$.
3. What are the differences between the estimated residuals $\hat\varepsilon_{1i}$ and $\hat\varepsilon_{2i}$ from the two regressions? Do their signs differ?

Concept: Two Equivalent Parameterizations of Group Means

An intercept plus one group dummy and two exhaustive group dummies without an intercept generate the same fitted values.

Let

Step-by-step Solution:

1. Intercept and one dummy

For $F_i=0$, $\widehat Y_i=\hat\alpha$, so

For $F_i=1$, $\widehat Y_i=\hat\alpha+\hat\beta=\bar Y_F$, so

2. Two dummies without an intercept

For the $M$ group, $M_i=1$ and $F_i=0$, so

For the $F$ group, $M_i=0$ and $F_i=1$, so

The coefficient mapping is

3. Residuals

Both specifications predict $\bar Y_M$ for the $M$ group and $\bar Y_F$ for the $F$ group. Therefore,

and

The residuals have the same values and signs in the two regressions.

3. OLS estimates of a simple regression model are $\hat\alpha=0$ and $\hat\beta=1/3$. You estimate the regression again after mean-centering both variables—that is, subtracting each variable’s sample mean.

1. How does mean-centering affect the coefficient estimates?
2. How does mean-centering affect $R^2$?

Concept: Mean-Centering

Define

Centering changes the origin but not the covariance between $X$ and $Y$ or the variance of $X$ around its mean.

Step-by-step Solution:

The centered slope is

Therefore,

Since $\bar X^c=\bar Y^c=0$,

Thus,

The fitted deviations and residuals are unchanged. The residual sum of squares and the centered total sum of squares are therefore unchanged, so

Answer: Mean-centering leaves the slope at $1/3$, gives an intercept of zero, and leaves $R^2$ unchanged.

Exercises

4. Assume that $X$ is stochastic and that $\hat\beta$ estimates $\beta$ in the model

Provide a formal proof that, under suitable assumptions, $\hat\beta$ is unbiased.**

Concept: Conditioning on the Regressor Sample

When $X$ is stochastic, condition on the observed values $\mathbf X=(X_1,\ldots,X_N)$. Under zero conditional mean, the error component of the OLS estimator has conditional expectation zero.

Assumptions:

1. $Y_i=\alpha+\beta X_i+\varepsilon_i$ is correctly specified.
2. $S_{XX}=\sum_{i=1}^{N}(X_i-\bar X)^2>0$.
3. $E(\varepsilon_i\mid\mathbf X)=0$ for every $i$.
4. The required expectations are finite.

Homoskedasticity and normality are not required for unbiasedness.

Proof:

The OLS slope can be written as

Substitute the model:

because

and

Conditional on $\mathbf X$,

By the law of iterated expectations,

Therefore,

5. Assume that $X$ is stochastic and that $\hat{\boldsymbol\varepsilon}$ is the OLS residual vector associated with

Provide a formal proof that

is an unbiased estimator of the error variance.**

Concept: Residual Degrees of Freedom

A simple regression with an intercept estimates two parameters. Under a conditionally spherical error covariance matrix, the expected residual sum of squares is $(N-2)\sigma^2$.

Let

and assume

1. $\operatorname{rank}(\mathbf Q)=2$;
2. $E(\boldsymbol\varepsilon\mid\mathbf Q)=\mathbf0$;
3. $\operatorname{Var}(\boldsymbol\varepsilon\mid\mathbf Q)=\sigma^2\mathbf I_N$.

Normality is not required.

Proof:

Write the model as

where $\boldsymbol\theta=(\alpha,\beta)’$. Define

and

The OLS residual vector is

because $\mathbf M\mathbf Q=\mathbf0$.

Since $\mathbf M$ is symmetric and idempotent,

Conditional on $\mathbf Q$,

The projection matrix $\mathbf H$ has rank $2$. Since an idempotent matrix has trace equal to its rank,

Therefore,

Dividing by $N-2$ gives

Applying iterated expectations,

Thus,

is an unbiased estimator of the common error variance.