4. (Standard Errors) For the following scenarios, we estimate the model using OLS. Explain whether the coventional standard error for the OLS estimator S (the one derived under the classic linear model assumptions) is valid or not. In your explanation, show necessary work to justify your conclusion. If the conventional standard error is not valid, explain what standard errors should be reported. (a) We want to examine the relation between consumption and income: con; = Po + Princi + ui. con is the annual consumption level, and inc is the annual income level. We observe that in the data, comparing to low income individuals, higher income individuals have a larger variance in their consumption level. (5 points) (b) We want to evaluate the effectiveness of a national subsidy program targeting low-income indi- viduals. The program gives low-income households tax reduction when they start small business. The tax reduction amount varies at the province level. We collect household-level information and estimate the following model: yip = Bo + Bisp + B2-Xip + tip. Uip is the annual income level five years after the program for household i living in province p, sp is the subsidy level, Xi represent other control varaibles. Besides, we believe uip = Up + nip: where up is a random component specific to province p, and nip is iid over i and p. (5 points) (c) We are interested in how one's risk aversion level is associated with their decision to buy life insurance: ins, = Bu + Birski + wi. ins is binary and equals one if individual buys insurance, rak is the risk aversion level. (5 points) (d) Consider the following time series regression model: It = Bo + Bist + Ct. ut = put-1 + vt, PE (0, 1). where r, and of are i.i.d. over t and are independent of each other. Assume E[w] = 0, E[v;] = 62. (10 points)