AAEC 4302 STATISTICAL METHODS IN AGRICULTURAL RESEARCH

1
AAEC 4302 STATISTICAL METHODS IN AGRICULTURAL
RESEARCH

• Chapter 7(7.1 7.2) Theory and Application of
  the Multiple Regression Model

2
Introduction

• The multiple regression model aims to and must
  include all of the independent variables X1, X2,
  X3, , Xk that are believed to affect Y
• Their values are taken as given It is critical
  that, although X1, X2, X3, , Xk are believed to
  affect Y, Y does not affect the values taken by
  them
• The multiple linear regression model is given by
• Yi ß0 ß1X1i ß2X2i ß3X3i ßkXki
  ui
• where i1,,n represents the observations, k is
  the total number of independent variables in the
  model, ß0, ß1,, ßk are the parameters to be
  estimated and ui is the disturbance term, with
  the same properties as in the simple regression
  model

3
Example
4
The Model

• In our example we have a time series data, k is
  five and i is twenty one.
• The model to be estimated, therefore, is
• Yi ß0 ß1X1i ß2X2i ß3X3i ß4X4i ui
• As before
• E Yi ß0 ß1X1i ß2X2i ß3X3i ßkXki
• Yi EYi ui , the systematic (explainable)
  and
• unsystematic (random) components of Yi
• And the corresponding prediction of Yi
• Yi ß0 ß1X1i ß2X2i ß3X3i ß4X4i

5
Model Estimation

• Also as before, the parameters of the multiple
  regression model (ßo, ß1, ß2, ß3, ß4) are
  estimated by minimizing SSR
• SSR ?ei2 ?(Yi- ß0 - ß1X1i - ß2X2i - ß3X3i -
  ß4X4i )2
• As before, the formulas to estimate the
  regression model parameters that would make the
  SSR as small as possible

n
n





i1
i1
6
Y
X2
Regression surface (plane) EY ßoß1X1ß2X2
Ui
X2 slope Measured by ß2
ßo
X1 slope measured by ß1
X1
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Model Estimation

• In general, only a model that is estimated
  including all of the (independent) variables that
  affected the values taken by Y in the sample will
  produce correct parameter estimates
• Only then the formulas for estimating these
  parameters would be unbiased

8
Model Estimation
9
Interpretation of the Coefficients

• The intercept ßo estimates the value of Y when
  all of the independent variables in the model
  take a value of zero which may not be
  empirically relevant or even correct in some
  cases.
• In our example ßo , is 144.94, which means that
  if
• Yi 144.94ß1(0)ß2(0) ß3(0)ß4(0)
• All the independent variables take the value of
  zero (price of beef is zero cents/lb, price of
  chicken is zero cents/lb, price of pork is zero
  cents/lb, and the income for US population is
  zero dollars/ per year, then the estimated beef
  consumption will be 144.94 lbs/year).

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Interpretation of the Coefficients

• In a strictly linear model, ß1, ß2,..., ßk are
  slopes of coefficients that measure the unit
  change in Y when the corresponding X (X1, X2,...,
  Xk) changes by one unit and the values of all of
  the other independent variables remain constant
  at any given level (it does not matter which)
• Ceteris paribus (other things being equal)

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Interpretation of the Coefficients

• In our example
• ß1 -0.00291. That means, if the price of beef
  increases by one cent/lb then the beef
  consumption will decrease by 0.00291 pounds per
  year, ceteris paribus
• ß2 -0.116. That means, if the price of chicken
  increases by one cent/lb then the beef
  consumption will decrease by 0.116 pounds per
  year, ceteris paribus (Does this result makes
  sense?)

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Interpretation of the Coefficients

• In our example
• ß3 0.3413. That means, if the price of pork
  increases by one cent/lb then the beef
  consumption will increase by 0.3413 pounds per
  year, ceteris paribus (beef and pork are
  substitutes).
• ß4 0.3121. That means, if the US income
  increases by one dollar per year then beef
  consumption will increase by 0.3121 pounds per
  year, ceteris paribus

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The Models Goodness of Fit

• The same key measure of goodness of fit is used
  in the case of the multiple regression model
• R2 1 - ?ei2 / ? (Yi-Y)2
• The only difference is in the calculation of the
  eis, which now equal
• ei Yi-ßo-ß1X1i-ß2X2i -ß3X3i--ßkXki

n
n
i1
i1





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The Models Goodness of Fit

• A disadvantage of the regular R2 as a measure of
  a models goodness of fit is that it always
  increases in value as independent variables are
  added into the model, even if those variables
  cant be statistically shown to affect Y
• This happens because, when estimating the models
  coefficients by OLS, any new independent variable
  would likely allow for a smaller SSR

15
The Models Goodness of Fit

• The adjusted or corrected R2 denoted by R2 is
  better measure to assess whether the adding of an
  independent variable likely increases the ability
  of the model to predict Y
• R2 1 ? ?ei2/(n-k-1)/?(Yi-Y)2/(n-1)
• The R2 is always less than the R2, unless the R2
  1
• Adjusted R2 lacks the same straightforward
  interpretation as the regular R2 under unusual
  circumstances, it can even be negative

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The Specification Question

• Any variable that is suspected to directly affect
  Y, and that did not hold a constant value
  throughout the sample, should be included in the
  model
• Excluding such a variable would likely cause the
  estimates of the remaining parameters to be
  incorrect i.e. the formulas for estimating
  those parameters would be biased
• The consequences of including irrelevant
  variables in the model are less serious if in
  doubt, this is preferred

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The Earnings Function

• Multiple regression model of the earnings
• EARNSi ß0 ß1EDi ß2EXPi ui
• Cross-section data set, 100 observations
• EARNSi -6.179 0.978EDi 0.124EXPi
• R2 0.315 SER4.288

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The Earnings Function

• Estimated coefficient ß2 0.124
• An additional year of experience increases
  earnings by 0.124 thousand dollars (124) ceteris
  paribus (holding constant the level of education)
• The effect on earnings of the difference in years
  of experience between the youngest and oldest men
  in our sample
• ?X2 30 0.124 30 3.720 thou difference
  in annual earnings

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The Earnings Function

• Estimated coefficient ß1 0.978
• This means that each additional year of schooling
  increases earnings by 0.978 thousand dollars
  (978), ceteris paribus (holding constant the
  level of experience)
• In simple regression ß1 0.797

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The Earnings Function

• What would be the total economic effect of going
  to school for two more years ?ED2 and ?EXP
  -2
• ?EARNS (0.978) 2 (0.124)(-2) 1.708
  thousand dollars per year
• Model results are used for predicting earnings
• EARNS -6.179 (0.978)(16) (0.124)(5)
  10.089