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- In a typical multiple linear regression model where x1 and x2 are non-random regressors, the expected value of the response variable y given x1 and x2 is denoted by E(y | 2,, X2). Build a multiple linear regression model for E (y | *,, *2) such that the value of E(y | x1, X2) may change as the value of x2 changes but the change in the value of E(y | X1, X2) may differ in the value of x1 . How can such a potential difference be tested and estimated statistically?Consider the following simple linear regression model: y = β0 + β1x + u. Using a sample of n observations on x and y, you estimate the model by OLS and obtain the estimates βˆ 0, βˆ 1, and the R-squared of the regression, R2 . Then you scale this sample by a factor of 100, obtain a new sample {xi/100; yi/100} for i = 1, . . . , n, re-estimate the model by OLS, and denote the new coefficient estimates by β˜ 0, β˜ 1, and the new R-squared of the regression by R˜2 . a) Give the expression of β˜ 1 in terms of βˆ 1, and justify your answer.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?
- Consider the following regression model Yt = β0 + β1 Ut + β2 Vt + β3 Wt + β4Xt + ∈t , where U, V, W, X and Y are economic variables observed from t = 1, . . . , 75, β0 , . . . , β4 are the model parameters and ∈t is the random disturbance term satisfying the classical assumptions. Ordinary Least Squares (OLS) is used to estimate the parameters, producing the following estimated model: Yt = 1.115 + 0.790*Ut − 0.327*Vt + 0.763*Wt + 0.456*Xt (0.405) (0.178) (0.088) (0.274) (0.017) where standard errors are given in parentheses, the R-squared = 0.941, the Durbin-Watson statistic is DW = 1.907 and the residual sum of squares is RSS = 0.0757. In answering this question, use the 5% level of significance for any hypothesis tests that you are asked to perform, state clearly the null and al- ternative hypotheses that you are testing, the test statistics that you are using and interpret the decisions that you make.…Consider the following datasets: X1=2,8,4 X2= 0.4, 7.10, 3.2 Y= 2.6, 9.2, 5.3 Statistically regress Y on X1 and X2, i.e. find a regression equation in which output variable is Y and input variable is X1 and X2. Show first two iterations of Gradient Descent method to solve part a. Initialize slopes and intercept at 0 value.Let yt = φyt−1 + et with et ∼ WN(0,σ2) and |φ| < 1. Consider the over-differenced process wt = (1 − L)yt.(i) What is the model followed by wt? (ii) Is wt invertible? (iii) Obtain V [wt] and compare its magnitude with V [yt] and hence comment on the impact of over-differencing on the variance of a stationary process.
- Consider the following regression Yi = βXi + ui. Show that (a) the OLS estimator of β is βˆ = (Σ(XiYi))/(Σ(Xi^2)) (b) (Σ(uˆi)/)n = 1/n(Σ(Yi − Yˆi))=/0Consider the following population linear regression model of individual food expenditure: Y = 50 + 0.5X + u, where Y is weekly food expenditure in dollars, X is the individual’s age, and 50+0.5X is the population regression line. Suppose we generate artificial data for 3 individuals using this model. This artificial sample, which consists of 3 observations, is shown in the following table: Answer the following questions. Show your working. (a) What are the values of V1 and V4? (b) Suppose we know that in this artificial sample, the sample covariance between X and Y is 150, and the sample variance of X is 100. Compute the OLS regression line of the regression of Y on X. (Hint: Assume these summary statistics and the OLS regression line continue to hold in parts (c)-(e).) (c) What are the values of V5 and V7?1. Suppose that the sales of a company (Y) is regressed on advertising expenditure (x) and labor cost (z), and the estimated regression equation is Y = 5 + 0.5x + 0.7z + u (where u is the error term). Here, sales, advertising expenditure and labor cost are measured in million Tk. Standard error for the coefficient of x is 0.4, standard error for the coefficient of z is 0.01, and the sample size is 20. Based on this information, find out whether labor cost is a statistically significant variable using an appropriate statistical test.
- Question 1 Provide an algebraic proof that the least squares estimator is not consistent when Cov(x,e)=0 with the regression model y =B1+B2E(x)+e where E(e)=0So that E(y) = B1 + B2E(x)Find the critical values of Z for a two tail test at alpha α = 0.01 LOS.The birth lengths in cm (x) and birth weights in kg (y) of a sample of 50 newborn female babies are compared, yielding a correlation coefficient of r=0.578 and a linear regression equation of ŷ =−8.89+0.243x The babies all had lengths between 46.5 and 53.0 cm, and weights between 2.50 and 4.05 kg. Based on this, predict the birth weight of a newborn female baby with a birth length of 48.5 cm.