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Multivariate Regression

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:

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Multivariate Regression

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  1. 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

  2. Data • Time Horizon: Monthly 3/1930-10/1936 • Response Variables: • Y1≡ Price of Danish Butter (Inflation Adjusted) • Y2≡ Price of New Zealand Butter (Inflation Adjusted) • Predictor Variables: • X1≡ Danish Imports • X2≡ New Zealand/Australia Imports • X3≡ All Other Imports

  3. p Responses k Predictors n observations Multivariate Regression Model

  4. Least Squares Estimates Note: This assumes independence across months

  5. 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):

  6. Butter Price Example

  7. 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 H0: LbM = d for specific matrices L,M,d

  8. Matrix Set up for General Linear Tests

  9. Three Statistics Based on H and E

  10. Testing Relation Between Price and Quality (I)

  11. Testing Relation Between Price and Quality (II)

  12. 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

  13. Matrix Forms for H1:H4

  14. 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

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