Multivariate Regression British Butter Price and Quantities from

Multivariate Regression British Butter Price and Quantities from Denmark and New Zealand 1930 -1936 I. Hilfer (1938). “Differential Effect in the Butter Market, ” Econometrica, Vol. 6, #3, pp. 270 -284

Data • Time Horizon: Monthly 3/1930 -10/1936 • 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 (x 0 is used for intercept term):

Butter Price Example

Testing Hypotheses Regarding b • 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: Lb. 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 – H 1: Common Effects for each quantity / price, common intercepts – H 2: Common Effects for each quantity / price, different intercepts – H 3: Common Effects for quantities, different effects across prices – H 4: Differential Effects for quantities, common effects across prices

Matrix Forms for H 1: H 4

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