Back to House Prices… Our failure to reject the null hypothesis implies that the housing stock has no effect on prices – Note the phrase cannot reject This is not very plausible. Even if it is true maybe it is an effect of the bubble – Prices became divorced from their usual determinants Re-estimate the model for the pre bubble period and see if there is difference There seems to be a difference after 1997

Structural Break This is known as a structural break or a regime shift Implies that the coefficients may be different not just the variables So the conditional expectation function has a kink Can happen at a point in time or for a different group of observations

A Regime Shift Show three data points for illustration

House Prices

Estimating with Structural Break Stata command: regress … if condition regress price inc_pc hstock_pc if year<=1997 Source | SS df MS Number of obs = F( 2, 25) = Model | e e+09 Prob > F = Residual | e R-squared = Adj R-squared = Total | e Root MSE = price | Coef. Std. Err. t P>|t| [95% Conf. Interval] inc_pc | hstock_pc | _cons |

Interpreting Results We can reject the null that housing stock has no effect on price – How? Do the details yourself It has the correct sign i.e. as theory would suggest We can use this pre-bubble model to predict what prices would have been – Use predict command – Note this command predicts out of sample also

The Pre Bubble Model

Interpreting the Graph Historically house prices have been determined by per capita income and the per capita housing stock. We estimate the parameters of that relationship before 1997 If the relationship had remained constant after 1997 prices would have followed the red line They difference between the two is huge (100% at peak)

Interpreting the Graph So prices rose rapidly even as the stock of houses rose Note we are not saying that prices should have remained constant They should have risen in line with income But the rose by more If the parameters had remained the same then prices would have followed the red line

But the marginal effect of income and stock should have remained the same – Income effect rose (more positive) – Stock effect rose (less negative) The change in parameters is the structural break that constitutes the bubble Note: this definition of a bubble would not be universally accepted

How Low Will They Go? If we believe our model then the parameters from pre 1997 still hold This doesnt mean that prices will return to 1997 levels It does mean that prices will return to the level determined by todays income (and stock) using the pre-bubble coefficients The red line represents the future level of house prices This represents a decline of 53% from 2010 levels!!!

Quality of Prediction Before we get too confident in this (or any) model it is worthwhile to assess whether it is a good predictor. We measure this using the R2 (R-Squared) and the adjusted R2 Both of these measure how much of the variation in the Y variable is explained by all the X variables collectively

R2R2

R 2 = ESS/TSS It gives the proportion of the total variation in the Y variables that is explained by the model By all the x variables collectively It doesnt say anything about which of the X variables are responsible for what portion of the variance – Doesnt say if the individual coefficients are statistically significant (multicolinearity) – Doesnt say if the individual coefficients make economic sense If want to use the model to predict Y then we want R2 to be high – Model explained a lot of the historical variation so it should explain a high proportion of the future variation – i.e. make good predictions Doesnt automatically follow that high R2 model is better than lower R2 model

Adjusted R 2

Using either R 2