Mathematically, multiple linear regression is:

$$Y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + ... + \beta_k x_k + \epsilon$$

We know that unemployment cannot entirely explain housing prices. To get a clearer picture of what influences housing prices, we add and test different variables and analyze the regression results to see which combinations of predictor variables satisfy OLS assumptions, while remaining intuitively appealing from an economic perspective.

We arrive at a model that contains the following variables: fed_funds , consumer_price_index , long_interest_rate , and gross_domestic_product , in addition to our original predictor, total_unemployed .

Adding the new variables decreased the impact of total_unemployed on housing_price_index . total_unemployed ‘s impact is now more unpredictable (standard error increased from 0.41 to 2.399), and, since the p-value is higher (from 0 to 0.943), less likely to influence housing prices.

Although total_unemployed may be correlated with housing_price_index , our other predictors seem to capture more of the variation in housing prices. The real-world interconnectivity among our variables can't be encapsulated by a simple linear regression alone; a more robust model is required. This is why our multiple linear regression model's results change drastically when introducing new variables.

That all our newly introduced variables are statistically significant at the 5% threshold, and that our coefficients follow our assumptions, indicates that our multiple linear regression model is better than our simple linear model.

The code below sets up a multiple linear regression with our new predictor variables.