Reference no: EM132403224

Econometric Methods

Assignment

1. Let A be a symmetric (n × n)-matrix with spectral decomposition

A = Σⁿ_i=1 λ_iv_iv'_i ,

where λi and vi are the eigenvalues and eigenvectors, respectively, of A such that v1, . . . , vn form an orthonormal basis of Rn. Define

A⁻ = Σⁿ_i≠1 1/λ_iv_iv'_i , .

(a) Show that A⁻ is a generalized inverse of A, that is, AA⁻A = A.

(b) Verify that A⁻ is a symmetric and reflexive generalized inverse (i.e., (A⁻) 0 = A⁻ and A is a generalized inverse of A⁻ so that A⁻AA− = A⁻).

2. Let X be an (n × k)-matrix (not necessarily of full rank) and P = X(X'X) −X'.

(a) Explain why a generalized inverse (g-inverse) of X0X does exist.

(b) Show that P is invariant w.r.t. the choice of the g-inverse of X'X (that is, any choice of the g-inverse of X'X gives the same matrix P).

Hint: Verify first that P X = X holds independent of the choice of the g-inverse.

(c) Show that P is the orthogonal projection from Rⁿ onto R(X).

(d) Show that P is symmetric and idempotent.

Hint: Use Problem 1.

3. Show that for any (m × n)-matrix A, there exists a g-inverse A⁻.

Hint: Set A⁻ = (A'A)⁻ A'

(why does (A'A) − exist?).

4. Prove that an (n × m)-matrix A⁻ is a g-inverse of a given (m × n)-matrix A if, and only if, x = A⁻b is a solution to the system of linear equations Ax = b for all b ∈ R(A) ⊆ R m.

5. Let Y_i ∼ P_θ = N (θ, 1) i.i.d., i = 1, ..., n, and consider the null hypothesis H₀ : θ = θ₀. With the test statistic λ = √n(Y^¯ −θ₀) and its observed value λ_obs = √n(y¯−θ₀), the pvalue for the right-tailed test H₀ vs. H^r₁ : θ > θ₀ is given by p_r = p_r(y) = Pθ₀(λ > λ_obs).

(a) Define the p-values p_l and p for the left-tailed test (H₀ vs. H^l₁ : θ < θ₀) and the two-sided test (H₀ vs. H₁ : θ≠ θ₀), respectively. Represent the p-values p_r, p_l and p by means of the distribution function Φ of the standard normal distribution, and show that p = 2 min{p_l , p_r}.

(b) Invert the UMP α-test δ^∗ for testing H₀ vs. H^r₁ (which rejects H0 if and only if λ_obs > z^1−α , cp. slide 39 of Chapter 2 and Problem 11 of Assignment No. 1) in order to get a (1 − α)-confidence interval for θ. Is that confidence interval exact, and does it make sense to assess its properties by the (expected) length? Why is it reasonable to speak in this case about a (1 − α) (lower or upper?) confidence bound for θ?

6. Suppose it holds

y_i = α₁ + α₂x_i2 + α₃x_i3 + α₄z_i + α₅x_i2z_i + u_i with E(u_i |x_i2, x_i3, z_i) = 0,

but z_i is not observable [so that the parameters α = (α₁, . . . , α₅)' cannot be estimated by OLS]. Assume that zi depends linearly on the observable variables x_i2, x_i3 by

z_i = γ₁ + γ₂x_i2 + γ₃x_i3 + v_i , E(v_i |x_i2, x_i3) = 0.

(a) Show that the model can be rewritten as

y_i = β₁ + β₂x_i2 + β₃x_i3 + β₄x²_i2 + β₅x_i2x_i3 + ε_i with E(ε_i |x_i2, x_i3) = 0.

(b) Find the partial effects of x_i2 and x_i3 on E(y_i |x_i).

(c) Can the OLS estimator of β = (β₁, . . . , β₅)' be given in closed form? Explain briefly.

(d) Show that E(ε_i) = 0.

(e) What can you conclude about E(ε_i |x_i2, x_i3, x²_i2 , x_i2x_i3)?

(f) Show that ‘any’ function f(x_i2, x_i3), possibly vector-valued with E||f (x_i2, x_i3)ε_i||² < ∞, is uncorrelated with ε_i.

7. (a) Show that the OLS estimators of the slopes in a multiple linear model that contains an intercept are obtained by transforming the data to deviations from their means and then regressing the dependent variable y in deviation form on the explanatory variables, also in deviation form (without intercept). Explain how you can get the OLS estimator of the intercept from the slope estimators.

(b) Verify the bias formula on slide 37 of Chapter 3.

8. Show that the coefficient of determination, R² , for a linear regression model containing an intercept is invariant with respect to linear transformations of the dependent variable, whereas the uncentered coefficient of determination, R^˜2 , is not. That is, R² remains unchanged after changing the scale and units of yi by γ and α, respectively, leading to new observations

y^∗_i = α + γy_i (α, γ ∈ R, γ ≠ 0),

whereas a change of units of yi (by α, say) leads to a change of R^˜2 .

9. Consider the linear regression model with intercept

y = Xβ + ε, rk(X) = k.                                            ...........(1)

Extending this model by one regressor leads to the model

y = Xθ + zγ + u, rk(X, z) = k + 1,                                            ...........(2)

where z is an n-dimensional column vector. Let e and ˆu denote the vector of OLS residuals in models (1) and (2), respectively.

(a) Verify that ||uˆ||² = ||e||²−γˆ²||Mz||², where ˆγ denotes the OLS estimator of γ in model (2) and M = In − X(X'X)⁻¹X' is the residual maker in (1).

(b) Let R^¯2 (i) be the adjusted coefficient of determination in model (i), i = 1, 2. Use the result in part (a) to show that R^¯2(2) > R^¯2 (1) if and only if |t_z|² > 1, where tz denotes the t-statistic for testing H₀ : γ = 0 in model (2).

10. Verify that, under the assumptions of Section 2.2.5, the best linear prediction P^∗ (X) = α^∗ + x'β^∗ is also the best linear approximation to P(X) = E[Y |X], that is,

min_α,β E[P(X) − α − X'β]² = E[P(X) − P^∗(X)]² .

11. (a) Show that in the regression model

y_i = ( ^k∏_j=1 x^βj_ij ) ε_i with x_ij > 0 and E(ε_i |x_i) = 1 (i = 1, . . . , n)

the parameter β_j is exactly the elasticity of E(y_i |x_i) with respect to x_ij.

(b) Verify that the parameter β_j in the model y_i = exp(x'_iβ)ε_i with E(ε_i |x_i) = 1 is exactly the semielasticity of E(y_i |x_i) with respect to x_ij .

12. Read carefully the following statements about the normal linear model y = Xβ + ε, ε ∼ N (0, σ² I_n), where the (n × k)-matrix X has full rank k and β = (β₁, . . . , β_k)'. Are they true or false? Explain.

(a) Since the OLS estimator βˆ = (βˆ₁, . . . , βˆ_k) 0 is BLUE of β, ˆγ₁ = P^k_j=1 βˆ_j and ˆγ₂ = βˆ²₂ are also BLUEs for estimating γ₁ = P^k_j=1 β_j and γ₂ = β²₂ , respectively.

(b) The hypothesis that the OLS estimator is equal to zero can be tested by means of a t-test.

(c) If the absolute t-value of a coefficient is smaller than t^0.975_n−k , we accept the null hypothesis that the coefficient is zero with 95% confidence.

(d) If an explanatory variable has a significant impact on the dependent variable at the 5% level, then its impact is also significant at the 10% level.

(e) If an explanatory variable has a significant impact on the dependent variable at the level α and its estimated coefficient is positive, then it has also a significant positive impact at the level α.

(f) The smaller the p-value, the stronger the evidence against the null hypothesis provided by the data.

(g) The p-value is the probability that the null hypothesis is true.

13. Empirical example (you can use any of the following software: EViews, Stata, R). Consider the following linear regression model (LRM)

lwage = β₁ + β₂ · educ + β₃ · exper + ε             ............(3)

and use the data in “em hw2 lwage.raw” (or “em hw2 lwage.xls”) for this exercise to answer the following questions.

Variable name	Variable description
Lwage	Logarithm of hourly wage
Educ	Years of education
Exper	Work experience in years
Urban	Dummy for urban areas
Married	Dummy for married individuals
meduc	Mother's education
feduc	Father's education

(a) Estimate the linear regression model in (3) using OLS. Give an economic interpretation of the estimated coefficients and comment on their significance. Does education and experience explain a lot of the variation in lwage?

(b) Extend equation (3) to allow the effect of educ to depend on the level of work experience and estimate the new model by OLS. Let θ be the semielasticity of wage w.r.t. education after 10 years of work experience. Estimate θ and test whether it is significantly different from zero at the 1% significance level.

(c) State [in the model of part (b)] the null hypothesis that the return to education does not depend on the level of experience. What do you think is an appropriate alternative?

(d) Extend the LRM in (3) in a way that allows lwage to differ across four groups of individuals: married and urban, married and rural, non-married and urban, non-married and rural. What is the estimated lwage differential between married urban and non-married urban? Is it significant at the 5% level?

(e) Extend the LRM in (3) by the variables urban, married, feduc and meduc and estimate the model again. Comment on differences compared to your answer from part (a).

(f) For the model considered in part (e), state the null hypothesis that mother’s and father’s education have jointly no impact on lwage. Perform a test at the 5% significance level.

(g) What would you assume about the error term ε to justify the OLS estimation and the tests carried out in the above models?

Attachment:- Em_ha2019_2.rar