Confidence Interval for Linear Regression

Question

I'm in an Econometrics class and we just finished taking our final exams. In the exam there was one particular question regarding confidence intervals that I can't figure out. I'm sure there's a simple answer but it's going to drive me crazy not being able to figure it out. Any insight/direction would be greatly appreciated!

The Question

A linear regression has been estimated for predicting the value of a house given several factors describing the house.

$$\widehat price = \underset{(8,452)}{245,595} + \underset{(0.23)}{1.20(sqrft - 2200)} + \underset{(824)}{2465(bdrms - 4)} + \underset{(76)}{1256(location-1)}$$

Where:

$sqrft$ is the square footage of the house

$bdrms$ is the number of bedrooms in the house

$location$ is a binary variable indicating the home resides in a favorable location

Given the estimated model, calculate the 90% confidence interval for the estimated price of a house with 2200 square feet, 4 bedrooms, and resides in a favorable location.

Assume the relevant $t$ statistic is $1.65$.

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Given this question and my basic knowledge of statistics I naively assumed that given the specification of the house each coefficient would be multiplied by zero leaving us with only the intercept.

Assuming that, we would then calculate a confidence interval using the equation:

$$C.I. = \hat\beta \pm t \times se(\hat\beta)$$

In this particular case:

$$C.I. = 245,595\pm(1.65\times8,452)$$

However the resulting confidence interval was not a valid choice. So, my limited knowledge of stats now exhausted, I gave up and made my best SWAG :(

Answer 1

I believe what you need is actually a prediction interval since you are bounding it for one point, not for the mean.

In that case, you should adjust your model to the s.e. prediction, which should be

Answer 2

Your proposed CI equation,

$$C.I._{proposed} = \hat\beta \pm t \times se(\hat\beta)$$

Is problematic in that $ \hat\beta $ is a vector, not a scalar. If we multiply by the input vector $X$ (where $X=[1\ sqrft\ bdrms\ location]$ we get

$$C.I. = X\hat\beta \pm t \times X(se(\hat\beta))$$

which is a scalar, and is therefore at least plausible. This makes intuitive sense because we expect that uncertainty in the house price should be a function of the uncertainty in all the model parameters, contrary to your assumption that the uncertainty comes solely from that of the intercept.

Answer 3

If I understand your notation you describe the standard error of each of your coefficients, including the intercept, below their respective figures. To calculate the correct 90% CI around your estimate, first you have to correctly calculate your estimate. And, I think in your case it is calculated as follows: estimated price = 245,595 + 1.2(2,200) + 2,465(4) + 1,256(1). That's if I understand your location variable correctly. I don't really know how it works, so I just plugged in 1 (in absence of any information regarding this variable). Assuming my guestimated calculations are correct, next you would use your estimated price + 1.65 the Standard Error of the whole Regression (not the Intercept) to estimate the high range of your 90% CI. Finally, you would calculate the low range of your 90% CI by taking your estimate - 1.65 the Standard Error of the whole Regression. And, you are done.