Q: The error term is homoskedastic if a. var (εi | Xi = x) is constant...
Q: The error term is homoskedastic if
a. var (εi | Xi = x) is constant for i = 1,…, n.
b. var (εi | Xi = x) depends on x
c. Xi is normally distributed
d. there are no outliers
d. the value of Yi is changes in a constant way
Homework Answers
Option a.
It refers to a condition where in the variance of the residual term in regression model is constant. So with homoskedasticity, the regression model would be accurate as the error term would not get influenced by the variables.
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Q: The error term is homoskedastic if
a. var (εi | Xi = x) is constant...
The error term is homoskedastic if var(u_i│X_i=x) is constant for i=1,2,…,n var(u_i│X_i=x) depends on x. X_i...
The error term is homoskedastic if var(u_i│X_i=x) is constant for i=1,2,…,n var(u_i│X_i=x) depends on x. X_i is normally distributed there are no outliers
Which of the following is not one of the least squares assumptions used in Stock and...
Which of the following is not one of the least squares assumptions used in Stock and Watson to show that the OLS estimators are unbiased and consistent and have approximately a normal distribution in large samples? 1) large outliers are unlikely 2) the error term is homoskedastic, i.e., Var(ui ∣ X=x) does not depend on x 3) the sample (Xi,Yi),i=1,…,n constitutes an i.i.d. random sample from the population joint distribution of X and Y 4) the conditional mean of the...
(a) If var[X o2 for each Xi (i = 1,... ,n), find the variance of X = ( Xi)/n. (b) Let the continuous random variable Y...
(a) If var[X o2 for each Xi (i = 1,... ,n), find the variance of X = ( Xi)/n. (b) Let the continuous random variable Y have the moment generating function My (t) i. Show that the moment generating function of Z = aY b is e*My(at) for non-zero constants a and b ii. Use the result to write down the moment generating function of W 1- 2X if X Gamma(a, B)
(a) If var[X o2 for each Xi (i...
3. Let Xi, . . . , Xn be iid randoln variables with mean μ and...
3. Let Xi, . . . , Xn be iid randoln variables with mean μ and variance σ2. Let, X denote the sample mean and V-Σ, (X,-X)2. (a) Derive the expected values of X and V. (b) Further suppose that Xi,-.,X, are normally distributed. Let Anxn ((a)) an orthogonal matrix whose first rOw 1S be , ..*) and iet Y = AX, where Y (Yİ, ,%), ard X-(XI, , X.), are (column) vectors. (It is not necessary to know aij...
σ2). 6. Suppose X1, Yİ, X2, Y2, , Xn, Y, are independent rv's with Xi and Y both N(μ, All parameters μί, 1-1, ,n, and σ2 are unknown. For example, Xi and Yi muay be repeated measurements on a lab...
σ2). 6. Suppose X1, Yİ, X2, Y2, , Xn, Y, are independent rv's with Xi and Y both N(μ, All parameters μί, 1-1, ,n, and σ2 are unknown. For example, Xi and Yi muay be repeated measurements on a laboratory specimen from the ith individual, with μί representing the amount of some antigen in the specimen; the measuring instrument is inaccurate, with normally distributed errors with constant variability. Let Z, X/V2. (a) Consider the estimate σ2- (b) Show that the...
Consider the following information regarding a response variable y and an explanatory variable x. x⎯⎯=8∑(xi−x⎯⎯)2=696 ∑(xi−x⎯⎯)(yi−y⎯⎯)=−346x¯=8∑xi-x¯2=696...
Consider the following information regarding a response variable y and an explanatory variable x. x⎯⎯=8∑(xi−x⎯⎯)2=696 ∑(xi−x⎯⎯)(yi−y⎯⎯)=−346x¯=8∑xi-x¯2=696 ∑xi-x¯yi-y¯=-346 y⎯⎯=8Σ(yi−y⎯⎯)2=185Σ(yi−yˆ)2=13n=25y¯= 8 Σ(yi− y¯)2 = 185 Σ(yi− y^)2=13n= 25 a. Calculate b0 and b1.b. What is the sample regression equation? Predict y if x equals 10. c. Calculate the standard error of the estimate. d. Calculate and interpret the coefficient of determination.
4. Suppose X1, . . . ,X, are independent, normally distributed with mean E(Xi) and variance...
4. Suppose X1, . . . ,X, are independent, normally distributed with mean E(Xi) and variance Var(X)-σί. Let Żi-(X,-μ.)/oi so that Zi , . . . , Ζ,, are independent and each has a N(0, 1) distribution. Show that LZhas a x2 distribution. Hint: Use the fact that each Z has a xî distribution i naS
7. Show that σ2 E(X-0 and Var(X if X1, . . . , Xn are independent...
7. Show that σ2 E(X-0 and Var(X if X1, . . . , Xn are independent and identically distributed with E(Xi) = 0 and E(X2) = σ2 for i = 1,-.. , n
Exercise 7. Let Xi, X2, . . . be independent, identically distributed rundorn variables uithEX and Var(X) 9, and let Yǐ = Xi/2. We also define Tn and An to be the sum and the sample mean, respectivel...
Exercise 7. Let Xi, X2, . . . be independent, identically distributed rundorn variables uithEX and Var(X) 9, and let Yǐ = Xi/2. We also define Tn and An to be the sum and the sample mean, respectively, of the random variablesy, ,Y,- 1) Evaluate the mean and variance of Yn, T,, and A (2) Does Yn converge in probability? If so, to what value? 3) Does Tn converge in probability? If so, to what value? (4) Does An converge...
Log-lin model in simple regression: yi=B0.B1Xi.eui take natural log of both sides lnyi=lnB0 + (lnB1)xi +...
Log-lin model in simple regression:
yi=B0.B1Xi.eui
take natural log of both sides
lnyi=lnB0 + (lnB1)xi +
ui . Take lnyi = wi,
lnB0= B*0 and
lnB1=B*1 so the former equation
equals to wi= B*0 +
B*1 + ui. It is said that, when
you increase x by 1 unit, you expect B*1 %
change in y.
B*1
=(dyi/yi) / dxi =
= (((yi-yi-1)/yi) /
(xi- xi-1))*100 . question is about the
numerator, (yi-yi-1)/yi)*100:
1)It is said that the numerator reflects...
(a) If var[X o2 for each Xi (i = 1,... ,n), find the variance of X = ( Xi)/n. (b) Let the continuous random variable Y have the moment generating function My (t) i. Show that the moment generating function of Z = aY b is e*My(at) for non-zero constants a and b ii. Use the result to write down the moment generating function of W 1- 2X if X Gamma(a, B)
(a) If var[X o2 for each Xi (i...
3. Let Xi, . . . , Xn be iid randoln variables with mean μ and variance σ2. Let, X denote the sample mean and V-Σ, (X,-X)2. (a) Derive the expected values of X and V. (b) Further suppose that Xi,-.,X, are normally distributed. Let Anxn ((a)) an orthogonal matrix whose first rOw 1S be , ..*) and iet Y = AX, where Y (Yİ, ,%), ard X-(XI, , X.), are (column) vectors. (It is not necessary to know aij...
σ2). 6. Suppose X1, Yİ, X2, Y2, , Xn, Y, are independent rv's with Xi and Y both N(μ, All parameters μί, 1-1, ,n, and σ2 are unknown. For example, Xi and Yi muay be repeated measurements on a laboratory specimen from the ith individual, with μί representing the amount of some antigen in the specimen; the measuring instrument is inaccurate, with normally distributed errors with constant variability. Let Z, X/V2. (a) Consider the estimate σ2- (b) Show that the...
4. Suppose X1, . . . ,X, are independent, normally distributed with mean E(Xi) and variance Var(X)-σί. Let Żi-(X,-μ.)/oi so that Zi , . . . , Ζ,, are independent and each has a N(0, 1) distribution. Show that LZhas a x2 distribution. Hint: Use the fact that each Z has a xî distribution i naS
7. Show that σ2 E(X-0 and Var(X if X1, . . . , Xn are independent and identically distributed with E(Xi) = 0 and E(X2) = σ2 for i = 1,-.. , n
Exercise 7. Let Xi, X2, . . . be independent, identically distributed rundorn variables uithEX and Var(X) 9, and let Yǐ = Xi/2. We also define Tn and An to be the sum and the sample mean, respectively, of the random variablesy, ,Y,- 1) Evaluate the mean and variance of Yn, T,, and A (2) Does Yn converge in probability? If so, to what value? 3) Does Tn converge in probability? If so, to what value? (4) Does An converge...
Log-lin model in simple regression:
yi=B0.B1Xi.eui
take natural log of both sides
lnyi=lnB0 + (lnB1)xi +
ui . Take lnyi = wi,
lnB0= B*0 and
lnB1=B*1 so the former equation
equals to wi= B*0 +
B*1 + ui. It is said that, when
you increase x by 1 unit, you expect B*1 %
change in y.
B*1
=(dyi/yi) / dxi =
= (((yi-yi-1)/yi) /
(xi- xi-1))*100 . question is about the
numerator, (yi-yi-1)/yi)*100:
1)It is said that the numerator reflects...
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