AAEC 4302 ADVANCED STATISTICAL METHODS IN AGRICULTURAL RESEARCH

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AAEC 4302ADVANCED STATISTICAL METHODS IN
AGRICULTURAL RESEARCH

• Chapter 1.

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Introduction

• Econometrics involves special statistical methods
  that are most suitable for analyzing economic
  data/relations
• Linear regression is a primary tool for empirical
  economic and biological analyses

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Linear Regression Analysis
p 2

• The first step in a linear regression analysis is
  to state a behavioral relation based on economic
  or biological theories or plain reasoning
• The second step is to state this relation as a
  mathematical equation

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Example of a Linear Regression Model

• U is the error term takes into account other
  factors that affect the dependent variable Y,
  such as
• Individually unimportant variables
• Error in the measurement of Y
• Pure chance
• The model is an abstraction from reality

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Example of a Linear Regression Model

• A model seeks to capture the essentials of the
  biophysical or economic process under analysis
• A key assumption when using linear regression is
  that the model is specified correctly
• Not all equations in empirical economics are
  structural equations

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Example of a Linear Regression Model

• Suppose that the estimates for ß0 and ß1 are
  ( and ,
  therefore the estimated model is
• Y 624.3 25.1X1
• Y could be cotton production in lbs/acre and X
  could be water applied, inches/growing season

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Example of a Linear Regression Model

• One must recognize that the predictions made by a
  regression model will never be totally precise
• Due to the error term affecting the true
  (population) model
• Due to the fact that the parameters of that model
  (B0 and B1) are unknown, and have to be estimated
  using regression analysis

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An Example of an Economic Model
Y 624.3 25.1X1

• The estimated 25.1 value for ß1 indicates that if
  irrigation applied increases by 1 inch per
  season, production will increase by 25.1
  pounds/acre
• However, 25.1 is only an estimate of the true but
  unknown value of ß1 and, therefore, it is subject
  to estimation error
• How confident can one be on this conclusion?

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Brief Review of Functions and Graphs

• A function is a mathematical relation that
  associates a single value of the variable Y with
  each value of the variable X, in general form
• Y f(X)
• The equation of a line is given by
• Y b0 b1X
• In a linear equation, ß0 is the intercept and
  measures the value of Y when X 0

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Brief Review of Functions and Graphs

• ß1 is the slope of the line, which measures the
  unit change in Y when X changes by one unit
  (?Y/?X)

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Brief Review of Functions and Graphs

• ß1 is the slope of the line, which measures the
  unit change in Y when X changes by one unit
  (?Y/?X)
• If ß1 is positive (negative) the line slopes
  upward (downward) from left to right the larger
  ß1 (in absolute value), the steeper the line
• ß1 0 implies a horizontal line at Y ß0

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Brief Review of Functions and Graphs

• Many functions are not straight lines (example
  Y10X0.5)
• Their slope can be viewed as the slope of the
  straight line drawn tangent to the curve at that
  point
• It is also interpreted as the ratio of the change
  in Y to a change in X that results from moving
  along the curve just a small distance from the
  original point

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Brief Review of Functions and Graphs
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Brief Review of Functions and Graphs

• Graph the following functions
• Y 650 40X X2
• Y 650 - 40X X2
• for x values from 0 to 40
• Find their derivatives (dY/dX)
• Evaluate derivatives at different points

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A Brief Review of Elasticity

• The elasticity is an alternative way to measure
  the response of Y to changes in X, which refers
  to proportional (i.e. percentage) changes instead
  of unit changes (recall slope?unit changes)
• Given a function Y f(X), its elasticity at a
  given point (Y, X) is measured by (and
  interpreted as) the percentage (i.e.
  proportional) change in Y (?Y/Y) divided by the
  percentage change in X (?X/X)

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A Brief Review of Elasticity

• If the changes are restricted to be small
• Elasticity We can calculates the elasticity
  of a function at any particular (Y, X) value.

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A Brief Review of Elasticity

• A linear function has a constant slope, but its
  elasticity varies throughout the function
• In general, both the slope and the elasticity may
  change along a non-linear function
• However, there is a special kind of non-linear
  function which elasticity (but not its slope) is
  constant throughout.