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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 $1.65$.


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 :(

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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

PI

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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.

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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.

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