How Low Can House Prices Fall?

(Quite a bit)

Learning Outcomes

1. Expand the regression model to allow for multiple X variables

2. Formalise the hypothesis test procedure using test statistics

3. Look at more general hypothesis testsa) Multiple coefficientsb) Inequality hypotheses

4. Formalise a procedure for using regression for prediction

House Prices

• We all know the story of financial crisis– House price bubble– Banks borrow abroad to finance bubble– Bubble bursts and banks cant get money bank– Bankrupt banks bailed out by state– State bankrupted and bailed out by Troika

• See nama.biz if interested

Two Questions

1. Could we have used econometrics to test for the presence of a bubble?

2. Now that the bubble is burst, can we use econometrics to say how far have prices to fall?

• Yes to both

Immediate relevance

• Should you buy or rent now?• How big is remaining hole in the banks?• What about other bubbles? China?

Look at data

• Look at aggregate macro time series data of Irish house prices in housing.dta

• Stata: line psecd• Certainly a rapid rise• Was it justified?

– Incomes and population were rising at the time– By enough?

• Econometrics can answer this question

010

0000

2000

0030

0000

4000

0050

0000

pric

e of

sec

ond

hand

hou

ses

dubl

in, d

oe

1970 1980 1990 2000 2010year

House Prices

How to Answer the 2 Q

• Make use of conditional expectation interpretation of a regression– Recall that regression line gives E(Y|X)

• So we will use OLS to give us the expected price of a house conditional on income, population etc

• Answers1. If the actual price is systematically above the expected

price we have evidence of a bubble2. After burst the price will fall to the conditional

expectation

Extending OLS to Many Xs

• We need to understand how OLS works when there are many independent (RHS) variables

• Recall: E(Y|X)=b1+b2X• Generalise to:• E(Y|X)=b0+b1X1i+ b2X2i+ ...+bkXki

• So the full model becomes:• Yi = b0+b1X1i+ b2X2i+ ...+bkXki + ui

Interpreting bk

• Each parameter 2 , 3 …k measures the isolated effect of x2 , x3 , xk on the dependant variable y

• Partial Regression coefficients.• In terms of calculus bk is a partial derivative• The effect of changing one variable while keep

all others constant

Interpreting OLS

• OLS still gives the best line• The only difference is that the “line” isn't a

line any more, it is a multi-dimensional hyper plane

• The actual data still deviates from the “line”• The “line” is still the conditional expectation• So if confused use the intuition from the single

RHS variable case

Intuition of single X case is still valid

Y E(Y|X)=1+2X

Y1 u1

Y3 u3

Y2 u2

1 X2 X1 X3 X

2

( | )E Y XX

: slope coefficient: Change in E(Y|X)

for a change in x.

Show three data points for illustration

Maths of OLS

• The formulae for OLS are much more complicated

• Really need matrix algebra to write them down• But idea is same

– Choose estimates b0 …bk to minimise the sum of squared deviations

• Computer does it for us with the regress command

A Preliminary Answer• As with many projects we first need an economic

model – Just like Keynes consumption function

• Our model will assert that real house prices are a function of per capita real income and the per capita housing stock

• Need to generate variables from the raw data Generate– real house prices:gen price=psecd/p*100– Real per capita income: gen inc_pc=gni/pop*1000– Real pc housing stock: gen hstock_pc=hstock/pop

OLS Estimates of Modelregress price inc_pc hstock_pc

Source | SS df MS Number of obs = 41-------------+------------------------------ F( 2, 38) = 210.52 Model | 6.7142e+11 2 3.3571e+11 Prob > F = 0.0000 Residual | 6.0598e+10 38 1.5947e+09 R-squared = 0.9172-------------+------------------------------ Adj R-squared = 0.9129 Total | 7.3202e+11 40 1.8301e+10 Root MSE = 39934

------------------------------------------------------------------------------ price | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- inc_pc | 16.15503 2.713043 5.95 0.000 10.66276 21.6473 hstock_pc | 124653.9 389266.5 0.32 0.751 -663374.9 912682.6 _cons | -190506.4 80474.78 -2.37 0.023 -353419.1 -27593.71------------------------------------------------------------------------------

Predicting Prices

• We can use this model to predict prices• Recall that OLS gives us an estimate of the

conditional expectation function• Put in the numbers

– E(Y|X)=b0+b1X1i+ b2X2i+ ...+bkXki

– E(Price|inc_pc, hstock_pc)= -190506.4 + 124653.9*hstock_pc+ 16.15503*inc_pc

Predicting Prices

• We can use generate command to create a variable with this conditional expectation– gen pred=-190506.4 + 124653.9*hstock_pc+

16.15503*inc_pc• So useful there is a special command

– predict pred• Compare the two on a graph

– line price pred year

Predicted vs Actual Prices0

1000

00

2000

00

3000

00

4000

00

5000

00

1970 1980 1990 2000 2010year

price pred

Interpretation

• The predicted line is our estimation of what we would expect the price to be given the level of income and the housing stock

• The actual price was systematically above this expected price from 2003

• This could be seen as evidence of a bubble• Note how it fits in to the discussion of the time which said that

prices went up but this was justified given the rise in incomes• We show that prices went up more than we would “expect”

for the observed increase in incomes• So we show that prices too high in a precise sense

More can be done

• We can test hypotheses about the variables to see if they have effects consistent with theory

• If the effects were different from our preconceived notions we may be wary of trusting the estimates– i.e. we got a fluke sample

Housing Stock Has no Effect?

1. Test the null hypothesis that housing stock has no effect on prices

H0: bH = 0 H1: bH ≠ 0

2. Calculate the distribution of bOLS assuming that H0 is true.

3. Find our estimate on the distribution4. What is the probability that our estimate would

have come from this distribution?5. Does this lead us to believe the null hypothesis?

95% of distribution

• Quite possible to get an estimate of 124653.9 if the true value is 0.0

• Note that this has nothing to do with the scale of the estimate. The estimate is a big number but it is not statistically different from zero.

• Calculate the probability– P(bOLS≥124653.9| bH=0.0)=– P(z ≥(124653.9-0)/(389266.5))– P(z ≥0.32)= 0.37

• Clearly this is much larger than usual threshold values of 5%,10% or 1%

• So we cannot reject the null hypothesis

Comments• We could reject the null if our threshold was 40%

– Seems very extreme– Think of criminal trial metaphor

• Cannot reject idea that effect of housing stock is zero even though the estimated effect is 120000!– Scale of coeff has NO impact on statistical significance

• Result does seem unlikely as contradicts theory– How to resolve this contradiction?– Look carefully at both theory and estimates– Sniff test!

• Can simplify test procedure

Test Statistics

• Clearly a large degree of commonality between our tests even though they were on different data

• So we can systematize things a little better• The key part of each test was calculating Z

)1,0(~)(

)(,~

Nse

Z

VarN

OLS

OLS

OLSOLS

• So now we only ever have to deal with one distribution, the “standard” normal

• Note also how the construction of Z explicitly removes the issue of scale– Stn err has same scale as coeff.

• Stream-lined Test procedure1. State Hypothesis2. Calculate Z assuming H0 is true

3. Now we can compare the calculated values of Z with the standard normal distribution

The Housing Stock Example

1. State the Hypothesis we want to testH0: bH = 0 H1: bH ≠ 0

2. Calculate the test statistic assuming that H0 is true.z =(124653.9-0)/(389266.5)=0.32

3. Find our estimate on the distribution– Either find the test statistic on the standard normal

distribution– Or compare with one of the traditional threshold (“critical”)

values: 2.58(1%), 1.96 (5%), 1.64(10%)

4. |Z|<all the critical values 5. So we cannot reject the null hypothesis

95% of distribution

Comment

• We will reject the idea that bH = 0.0 if there is overwhelming evidence that bH is bigger or smaller

• The evidence is our estimate (120000)• Is this big enough? It looks huge• But the standard error is huge also – so a very

wide distribution of estimates• Remove the scale from the problem by

calculating the test statistic: Z=0.32

Comment

• Is this big enough?• Traditionally 1.96 would be the “critical value”

because of 5% probability of |Z|>1.96 as fluke• “beyond reasonable doubt”• Free to decide for ourselves (p-value)

– p-value=Pr(|Z|>0.32)=0.75

Issues in Hypothesis Testing

• Test of significance• “t-test”• Rule of thumb • Significance level

Test of Significance

• A test of H0: b = 0 is given the special name of “test of significance”

• Test statistic is simpleZ=(bOLS – )/b se(bOLS)= bOLS/se(bOLS)Which is calculated by most statistical software

• Simple eyeball test of significance• Variable is or is not “statistically significant”• Not the same as economically significant

t-test• Strictly speaking the Z test is only valid when s, the

variance of u is known as it is used to calculate se(b)

• s will almost never be known and will have to be estimated

• When it is estimated the distribution of the estimator (and therefore the test statistic) is no longer normal

• Has a t-distribution with N-K degrees of freedom

t-test• N is number of observations• K is the number of variables• The precise shape of the t distribution

depends on N-K• Note the contrast with the normal• Fortunately t≈Z when N-K is large• Stata reports t-test for statistical significance

automatically

Rule of Thumb• Easy to “learn off” test procedure

– Form the test statistic– Reject hypothesis if test statistic>2 in absolute

terms– Useful for “eyeball” tests of significance

• Stata command “test” does the whole procedure automatically– But does “F-test”– This is the square of t-test

Significance Level

• Probability of a fluke• When we choose a critical value we choose a

significance level also:– 2.58(1%), 1.96 (5%), 1.64(10%)

• If we reject the null because |Z|>1.96, we say we reject the null at the 5% significance level.

• We acknowledge that there is a 5% chance that Z>1.96 even though the null is true

• This is type 1 error: Rejecting a true Null– Criminal trial: Convicting the innocent

• The test is set up make this as low as possible– i.e. reject only if overwhelming evidence

• Why not make it zero? Cant because would never reject any null– Criminal: always acquit

• This matters because setting up a hypothesis is setting up a procedure that is deliberately biased against rejecting

• Make sure that is what you want for your null