Multivariate Regression British Butter Price and Quantities from Denmark and New Zealand I. Hilfer (1938). “Differential Effect in the Butter Market,” Econometrica, Vol. 6, #3, pp

Data Time Horizon: Monthly 3/ /1936 Response Variables: –Y 1 ≡ Price of Danish Butter (Inflation Adjusted) –Y 2 ≡ Price of New Zealand Butter (Inflation Adjusted) Predictor Variables: –X 1 ≡ Danish Imports –X 2 ≡ New Zealand/Australia Imports –X 3 ≡ All Other Imports

Multivariate Regression Model p Responses k Predictors n observations

Least Squares Estimates Note: This assumes independence across months

Butter Price Example p=2 Response Variables (Danish, NZ Prices) k=3 Predictors (Danish, NZ, Other Imports) n=80 Months of Data First and last 4 months (x0 is used for intercept term):

Butter Price Example

Testing Hypotheses Regarding  Many times we have theories to be tested regarding regression coefficients The most basic is that none of the predictors are related to any of the responses Others may be that the regression coefficients for one or more predictor(s) is same for two or more responses Others may be that the effects of two or more predictors are the same for one or more response(s) Tests can be written in the form of H 0 : L  M = d for specific matrices L,M,d

Matrix Set up for General Linear Tests

Three Statistics Based on H and E

Testing Relation Between Price and Quality (I)

Testing Relation Between Price and Quality (II)

Testing for a Differential Effect Hypothesis: Price of Danish and NZ Butter is equally “effected” by quantities of each type –H1: Common Effects for each quantity / price, common intercepts –H2: Common Effects for each quantity / price, different intercepts –H3: Common Effects for quantities, different effects across prices –H4: Differential Effects for quantities, common effects across prices

Matrix Forms for H1:H4

H and E Matrices Note: F.05,6,75 =2.22, F.05,5,75 =2.34, F.05,4,150 =2.44, F.05,4,75 =2.49 All 4 hypotheses are rejected