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:– 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

Danish and NZ Butter Prices in Britain (3/30-10/36)

250.0

300.0

350.0

400.0

450.0

500.0

550.0

600.0

650.0

700.0

750.0

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81

month

Adj

uste

d Pr

ice

Denmark

NZ

Danish Prices vs Danish Imports

250

300

350

400

450

500

550

600

650

700

750

125 150 175 200 225 250 275 300

Danish Imports

Dan

ish

Pric

e

NZ Prices vs Danish Imports

250

300

350

400

450

500

550

600

650

700

750

125 150 175 200 225 250 275 300

Danish Imports

NZ

Pri

ces

Danish Prices vs NZ Imports

250

300

350

400

450

500

550

600

650

700

750

0 100 200 300 400 500 600 700

NZ Imports

Dan

ish

Pric

es

NZ Prices vs NZ Imports

250

300

350

400

450

500

550

600

650

700

750

0 100 200 300 400 500 600 700

NZ Imports

NZ

Pri

ces

Multivariate Regression Model

Σe

e

e

E

β

ββX

Y

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Y

EXβY

i'n

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1

'0

1

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n observations

Least Squares Estimates

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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):

Month(t) Y1(t) Y2(t) x0(t) x1(t) x2(t) x3(t)1 678.5 590 1 181 197 1252 593.2 546.6 1 175 194 1253 570.9 558.2 1 187 129 1894 599.3 577.9 1 260 61 279 77 487.1 448.2 1 206 170 44578 496.9 465.8 1 188 245 35279 483.6 417.8 1 190 299 38080 474 385.4 1 196 324 312

Butter Price ExampleX'X X'Y80 16066 26690 15647 40444.9 33857.5

16066 3319752 5139280 3293189 8059701 681262126690 5139280 10274494 4551068 13362790 1079429215647 3293189 4551068 3909851 7697519 6635274

INV(X'X) B-Hat1.27431 -0.004153183 -0.00108058 -0.00034378 980.1346 905.7373

-0.004153 1.84557E-05 2.2308E-06 -1.5207E-06 -1.12371 -0.89519-0.001081 2.2308E-06 1.456E-06 7.5069E-07 -0.48986 -0.69075-0.000344 -1.5207E-06 7.5069E-07 2.0386E-06 -0.43702 -0.36961

Y'Y Y'PY Sigma-Hat20967695 17594889.22 20674850.4 17342086.8 3853.213 3326.34817594889 14935051.93 17342086.8 14658806.1 3326.348 3634.813

)(37.0)(69.0)(90.074.904)(

)(44.0)(49.0)(12.113.980)(^

^

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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 H0: LM = d for specific matrices L,M,d

Matrix Set up for General Linear Tests

valuescritical eappropriat with statistics Compare :5 Step

statistics- eapproximat tos)statistic(st Convert te :4 Step

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responses,for is and predictorsfor matrix a is where

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Three Statistics Based on H and E

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Testing Relation Between Price and Quality (I)

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06-2.04E07-7.51E1052.1000344.

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Testing Relation Between Price and Quality (II)

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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

323122211211020140

32221231211130

32221231211120

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,:

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,:

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Matrix Forms for H1:H4

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H and E Matrices

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Hypothesis H E q1 q2 s m1 m2 r u t1 662762 63485.1 1 6 1 2 38 78 1 12 115751 63485.1 1 5 1 1.5 38 77.5 0.75 13 29741.2 -6159.4 292844 252802 2 2 2 -0.5 38 75.5 0.5 2 -6159.4 59334.3 252802 276246 4 649483 63485.1 1 4 1 1 38 77 0.5 1

Hypothesis FW df1 (all) df2W V FP df2P1 0.08742 132.236 6 76 0.91258 135.716 782 0.3542 27.7139 5 76 0.6458 28.4432 783 0.35824 25.1533 4 150 0.67817 20.2658 1584 0.08904 194.379 4 76 0.91096 199.494 78

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