(c) The Hypothesis function for Univariate Linear Regression is y = wo+Wx. The cost function associated with this hypothesis function h, parametrised by some wo and W1, is y) – h,(x(i)))?, where yli) and x) represent the output and input values of the ith training example. (i) Write out and provide an explanation for the general form of the hypothesis
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- A “Cobb–Douglas” production function relates production (Q) to factorsof production, capital (K), labor (L), and raw materials (M), and an errorterm u using the equation Q = λKβ1Lβ2Mβ3eu, where λ, β1, β2, and β3 areproduction parameters. Suppose that you have data on production and thefactors of production from a random sample of firms with the same Cobb–Douglas production function. How would you use regression analysis toestimate the production parameters?In the example of First-Order Model in Five Quantitative Independent Variables, the parameters, βi,i=1,…,5, are often called partial regression coefficients. What does "partial" mean here?In an instrumental variable regression model with one regressor, Xi, andone instrument, Zi, the regression of Xi onto Zi has R2 = 0.1 and n = 50.Is Zi a strong instrument? Would your answer change if R2 = 0.1 and n = 150?
- The Simple Linear Regression model is Y = b0 + b1*X1 + u and the Multiple Linear Regression model with k variables is: Y = b0 + b1*X1 + b2*X2 + ... + bk*Xk + u Y is the dependent variable, the X1, X2, ..., Xk are the explanatory variables, b0 is the intercept, b1, b2, ..., bk are the slope coefficients, and u is the error term, Yhat represents the OLS fitted values, uhat represent the OLS residuals, b0_hat represents the OLS estimated intercept, and b1_hat, b2_hat,..., bk_hat, represent the OLS estimated slope coefficients. QUESTION 7 In the MLR model, the assumption of ‘linearity in parameters’ is violated if: one of the slope coefficients appears as a power (e.g. Y = b0 + b1*(X1^b2) + b3*X2 + u) the model includes the reciprocal of a variable (e.g. 1/X1) the model includes a variable squared (e.g. X1^2) the model includes a variable in its logarithmic form (i.e. log(X1) ) QUESTION 8 In the MLR model, the assumption of 'no perfect collinearity'…The Simple Linear Regression model is Y = b0 + b1*X1 + u and the Multiple Linear Regression model with k variables is: Y = b0 + b1*X1 + b2*X2 + ... + bk*Xk + u Y is the dependent variable, the X1, X2, ..., Xk are the explanatory variables, b0 is the intercept, b1, b2, ..., bk are the slope coefficients, and u is the error term, Yhat represents the OLS fitted values, uhat represent the OLS residuals, b0_hat represents the OLS estimated intercept, and b1_hat, b2_hat,..., bk_hat, represent the OLS estimated slope coefficients. QUESTION 1 In the SLR model, suppose the dependent variable (Y) represents the quantity consumed of apples in a particular area in tones, and the explanatory variable (X1) is the average price of apples in that area in £. If this model is estimated by OLS, then the estimated slope b1_hat, represents: by how many tones consumption of apples will change, if the average price of apples increases by £1 the predicted change in the consumption of apples (in…The Simple Linear Regression model is Y = b0 + b1*X1 + u and the Multiple Linear Regression model with k variables is: Y = b0 + b1*X1 + b2*X2 + ... + bk*Xk + u Y is the dependent variable, the X1, X2, ..., Xk are the explanatory variables, b0 is the intercept, b1, b2, ..., bk are the slope coefficients, and u is the error term, Yhat represents the OLS fitted values, uhat represent the OLS residuals, b0_hat represents the OLS estimated intercept, and b1_hat, b2_hat,..., bk_hat, represent the OLS estimated slope coefficients. QUESTION 16 In a t-test, suppose a researcher sets the significance level at 0.5%. What does this mean? The probability that the null hypothesis is true is 0.5% The researcher would be rejecting the null hypothesis, only if the p-value is less than 0.5% The researcher would be rejecting the null hypothesis, if the t-statistic is higher than 0.5 It does not mean anything, because the significance level can only be set at 5% QUESTION 17 In an MLR…
- The Simple Linear Regression model is Y = b0 + b1*X1 + u and the Multiple Linear Regression model with k variables is: Y = b0 + b1*X1 + b2*X2 + ... + bk*Xk + u Y is the dependent variable, the X1, X2, ..., Xk are the explanatory variables, b0 is the intercept, b1, b2, ..., bk are the slope coefficients, and u is the error term, Yhat represents the OLS fitted values, uhat represent the OLS residuals, b0_hat represents the OLS estimated intercept, and b1_hat, b2_hat,..., bk_hat, represent the OLS estimated slope coefficients. QUESTION 28 Suppose your estimated MLR model is: Y_hat = -30 + 2*X1 + 10*X2 Suppose the standard error for the estimated coefficient associated with X2 is equal to 5. Now, suppose that for some reason we multiply X2 by 5 and we re-estimate the model using the rescaled explanatory variable. What will be the value of the estimated coefficient of X2 and its standard error? The estimated coefficient of X2 will be equal to 50 and its standard error will be…The Simple Linear Regression model is Y = b0 + b1*X1 + u and the Multiple Linear Regression model with k variables is: Y = b0 + b1*X1 + b2*X2 + ... + bk*Xk + u Y is the dependent variable, the X1, X2, ..., Xk are the explanatory variables, b0 is the intercept, b1, b2, ..., bk are the slope coefficients, and u is the error term, Yhat represents the OLS fitted values, uhat represent the OLS residuals, b0_hat represents the OLS estimated intercept, and b1_hat, b2_hat,..., bk_hat, represent the OLS estimated slope coefficients. QUESTION 13 In the MLR model, what do we mean by Heteroskedasticity? That the error term depends on the values of the explanatory variables That all the explanatory variables have different variance That the variance of the error term is a function of the explanatory variables That the variance of the error term is constant QUESTION 14 Suppose that in the model Y=b0+b1*X1+u, we add a variable that is correlated with both Y and X1. What will happen…The Simple Linear Regression model is Y = b0 + b1*X1 + u and the Multiple Linear Regression model with k variables is: Y = b0 + b1*X1 + b2*X2 + ... + bk*Xk + u Y is the dependent variable, the X1, X2, ..., Xk are the explanatory variables, b0 is the intercept, b1, b2, ..., bk are the slope coefficients, and u is the error term, Yhat represents the OLS fitted values, uhat represent the OLS residuals, b0_hat represents the OLS estimated intercept, and b1_hat, b2_hat,..., bk_hat, represent the OLS estimated slope coefficients. QUESTION 25 In the MLR model, with 150 observations and 5 explanatory variables, suppose you want to test the null hypothesis, H0: b3=0, b4=0 We also know that the RSS of the unrestricted model is 563, while the RSS of a model that excludes the variables associated with the 2 coefficients under the null hypothesis is 577. Given this information what is the F-statistic for this hypothesis? It is around -1.79 It is around 1.75 It is around -1.75 It is…
- The Simple Linear Regression model is Y = b0 + b1*X1 + u and the Multiple Linear Regression model with k variables is: Y = b0 + b1*X1 + b2*X2 + ... + bk*Xk + u Y is the dependent variable, the X1, X2, ..., Xk are the explanatory variables, b0 is the intercept, b1, b2, ..., bk are the slope coefficients, and u is the error term, Yhat represents the OLS fitted values, uhat represent the OLS residuals, b0_hat represents the OLS estimated intercept, and b1_hat, b2_hat,..., bk_hat, represent the OLS estimated slope coefficients. QUESTION 22 Suppose your estimated MLR model with two explanatory variables, X1 and log(X2), is: log(Y)_hat= 10 +0.5*X1 + 0.25*log(X2) Which of the following statements about the interpretation of the coefficient of log(X2) is correct? If X2 increases by 1%, Y is predicted to increase by approximately 25 units, holding X1 constant If X2 increases by 1%, Y is predicted to increase by approximately 0.25%, holding X1 constant If X2 increases by 1 unit, Y is…The Simple Linear Regression model is Y = b0 + b1*X1 + u and the Multiple Linear Regression model with k variables is: Y = b0 + b1*X1 + b2*X2 + ... + bk*Xk + u Y is the dependent variable, the X1, X2, ..., Xk are the explanatory variables, b0 is the intercept, b1, b2, ..., bk are the slope coefficients, and u is the error term, Yhat represents the OLS fitted values, uhat represent the OLS residuals, b0_hat represents the OLS estimated intercept, and b1_hat, b2_hat,..., bk_hat, represent the OLS estimated slope coefficients. QUESTION 4 Suppose we have an SLR model, where the dependent variable (Y) represents ‘how satisfied someone is with his/her life, from 0 to 100’ (the higher the value, the higher the satisfaction with life), and the explanatory variable (X1) represents ‘personal annual income in £1,000’. The estimated OLS regression line is: Yhat = 33.2 + 0.74*X1. According to this model, what is the predicted life satisfaction, for someone with…The Simple Linear Regression model is Y = b0 + b1*X1 + u and the Multiple Linear Regression model with k variables is: Y = b0 + b1*X1 + b2*X2 + ... + bk*Xk + u Y is the dependent variable, the X1, X2, ..., Xk are the explanatory variables, b0 is the intercept, b1, b2, ..., bk are the slope coefficients, and u is the error term, Yhat represents the OLS fitted values, uhat represent the OLS residuals, b0_hat represents the OLS estimated intercept, and b1_hat, b2_hat,..., bk_hat, represent the OLS estimated slope coefficients. QUESTION 14 Suppose that in the model Y=b0+b1*X1+u, we add a variable that is correlated with both Y and X1. What will happen to the standard error of the OLS estimator for b1? It will go up It will go down We cannot say with the provided information It will remain unchanged QUESTION 15 Suppose you have an MLR model that includes an intercept, with 150 observations and 11 variables. If assumptions MLR1-MLR6 hold, a t-statistic for any of the…
