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
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• The first step in a linear regression analysis is
  to state a behavioral relation based on economic
  or biological theories or plain reasoning
• This behavioral relation includes one dependent
  variable (Y) and several independent variables
  (Xs), which are believed to influence Y
• The second step is to state this relation as a
  mathematical equation
• Y B0 B1X1 B2X2 B3X3 U

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Linear Regression Analysis
Y B0 B1X1 B2X2 B3X3 U

• In this equation
• Y and X1, X2, and X3 are the dependent and the
  independent variables, respectively
• B0, B1, B2 and B3 are parameters, i.e. constant
  coefficients that describe the relations between
  Y and X1, X2 and X3

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Example of a Linear Regression Model

• Y B0 B1X1 B2X2 B3X3 U
• B0, B1, B2 and B3 are estimated using linear
  regression analysis
• U is an error or disturbance term, which
  recognizes that the relation between Y and X1,X2
  and X3 is not exact it takes into account other
  factors that affect the dependent variable Y

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Example of a Linear Regression Model

• Y B0 B1X1 B2X2 B3X3 U
• An example of a linear regression model that is
  useful for both, economic and biological analysis
  is a production function
• Estimating production functions is key to the
  economic analysis of production processes/ systems

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Example of a Linear Regression Model

• 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

• The parameters of the simple production function
  model Y B0 B1X1 U can be estimated using
  available data and linear regression techniques
• Suppose that the estimates for B0 and B1 are
  ( and ,
  therefore the estimated model is
• Y 624.3 25.1X1

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Example of a Linear Regression Model

• Y 624.3 25.1X1
• In the estimated model, Y is the value of Y (the
  dependent variable -production) that is expected
  or predicted to occur given a specific value of X
  (the independent or explanatory variable -input
  use)
• 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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Example of a Linear Regression Model

• In addition to making predictions, one may be
  interested in the signs and values of the
  parameters of an econometric model
• For example in the previous model, B1 gt 0
  indicates that applying irrigation water
  increases production

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An Example of an Economic Model
Y 624.3 25.1X1

• The estimated 25.1 value for B1 indicates that
  every inch of irrigation water applied increases
  production by 25.1 pounds/acre
• However, 25.1 is only an estimate of the true but
  unknown value of B1 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)
• In the graph of a function X is measured
  horizontally and Y vertically (i.e. Y is the
  height of the curve)
• The equation of a line is given by
• Y B0 B1X
• In a linear equation, B0 is the intercept and
  measures the value of Y when X 0 graphically
  it is the point where the line crosses the
  vertical (Y) axis

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Brief Review of Functions and Graphs

• B1 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

• B1 is the slope of the line, which measures the
  unit change in Y when X changes by one unit
  (?Y/?X)
• If B1 is positive (negative) the line slopes
  upward (downward) from left to right the larger
  B1 (in absolute value), the steeper the line
• B1 0 implies a horizontal line at Y B0

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Brief Review of Functions and Graphs

• Many functions are not straight lines (example
  Y10X0.5)
• Their slope is different at every point and 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

• In calculus, dy and dx are used instead of ?Y and
  ?X to signify that the changes are very small,
  thus
• Slope , the derivative of the function
  Y f(X) with respect to X

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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)

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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.