Question

1. After having estimated a regression model $LaTeX: y_{i} =\beta _{i} +\beta _{2} x_{i} +u_{i}$ , we wish to forecast the value of given a new observation $LaTeX: x^{\ast }$ . We can consider two types of predictions: forecasting the conditional mean of , or forecasting the actual value of . What is true about the variances of these predictions?

Var[conditional mean] > Var[actual value].

Var[conditional mean] = Var[actual value].

Var[conditional mean] < Var[actual value].

Var[conditional mean] may be larger than Var[actual value] in some data sets and smaller in others.

2. When working with the time series regression model $LaTeX: y_{t} =\beta _{i} +\beta _{2} x_{t} +u_{t}$ , in which case should one use HAC standard errors?

If we suspect that there is correlation between and .

If we suspect that there is correlation between and

If we suspect that there is correlation between and

Always, just to be on the safe side.

Answer 1

1. The forecast of the conditional mean is given by:

$E(y|x=x^*) = \beta_i + \beta_2 x^*$

and, the forecast of the actual value is given by:

$y = \beta_i + \beta_2 x^* + \mu_i$

The variances for the above two cases are given by $\beta_2^2 \sigma^2_x$ and , respectively.

oe μί

Therefore, the variance[conditional mean]<variance[actual value]

2. HAC is recommended for the cases when the error terms are serially correlated, and therefore, the correct answer is the third option, i.e. If we suspect that there is correlation between and
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