AAEC 4302 ADVANCED STATISTICAL METHODS IN AGRICULTURAL RESEARCH

1
AAEC 4302ADVANCED STATISTICAL METHODS IN
AGRICULTURAL RESEARCH

• Chapters 6 7
• Variables Model Specifications

2
Lagged Variables (6.3)

• In a standard simple regression model Yi depends
  on Xi
• When working with time series data, this means
  that the value of Y in time period t is (partly)
  explained by the value of X in time period t,
  i.e.
• ,
  where T is the last year of available data

3
Lagged Variables

• In many cases the value of Y in time period t is
  more likely explained by the value taken by X in
  the previous time period

4
Lagged Variables

• For example, a farmers current year investment
  decisions might be based on the previous year
  prices, since the current year prices are not
  known when making these decisions.
• Investment (Y) is dependent on last years prices
  X(t-1)
• Here, Y is said to be dependant on lagged values
  of the explanatory variable X

5
Lagged Variables

• In multiple regression models (i.e. models with
  more than one explanatory variable), it can be
  assumed that Y is affected by different lags of
  X

6
Lagged Variables

• The model
  can also be estimated using the OLS method
  (i.e. the previously developed formulas for
  calculating ( and )
• It is only necessary to rearrange the data in
  such a way that the value of Y at time period t
  coincides with the value of X at time period t-1

7
Lagged Variables
Without Lagged Variables
With Lagged Variables
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Lagged Variables
?
9
Lagged Variables

• Notice of course that the first observation on
  and the last observation on will not
  be used
• Multiple regression models can include lagged
  values of one or more of the independent
  variables

10
Lagged Variables
11
Lagged Variables
Suppose we want to estimate cotton acres planted
in the US (Y) as a function of the last 3 years
price of cotton lint (Xt), cents/lb.
What's the interpretation of 1.2 ?
It means that if the price of cotton lint three
years ago (t-3), changed by 1 cent per pound the
acres of planted cotton today (time, t) would
increase by 1.2 acres, while holding all the
other Xs constant.
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First Differences of a Variable (6.3)

• The first difference of a variable is its change
  in value from one time period to the next
• A common specification in time series models
  involves letting the dependent or the independent
  variable, or both, be specified in first
  differences

13
First Differences of a Variable

• First difference on Y
• First difference on X
• The only reason you do this is if you believe
  that it is not the previous year that affects Yt
  but the difference between the previous year and
  current year that affects Yt.

14
First Differences of a Variable

• An example of a first difference model is
• 
  , or
• In this case, the researcher has a reason to
  believe that it is the change in X which affects
  the value taken by Y, in a linear fashion

15
First Differences of a Variable
Suppose you wanted to estimate the function where
investment is a function of the change in GNP
(i.e. first difference).
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First Differences of a Variable
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Examples of First Difference Models

• In economics, the demand for durable goods could
  be more directly affected by the change in
  interest rates than by the interest rate level (a
  first difference in the independent variable)
• In forestry, deforestation (i.e. the change in
  the forest cover from one year to the next) could
  be more directly related to the price of wood
  than total forest cover (a first difference in
  the dependent variable)

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Examples of First Difference Models

• Notice that one observation is always lost when
  using first difference models

19
Alternative Model Specifications (6.3)

• The regression models studied so far assume that
  the relationships between Y and the independent
  variables are linear
• However, in many cases theory or experience
  indicates some of the individual Y-Xj
  relationships are likely non-linear where Xj
  represents any of the independent variables in
  the model

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Alternative Model Specifications

• Non-Linear Specifications
• Simplifying a model from a fitted plane to a
  fitted line, we multiply then mean of X2 times
  and add it to the intercept ( )

holding X2 at the mean
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Alternative Model Specifications
Y

or
X1
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Alternative Model Specifications

• Non-Linear Specifications
• Simplifying a model from a fitted plane to a
  fitted line, we multiply then mean of X1 times
  and add it to the intercept ( )

holding X1 at the mean
23
Alternative Model Specifications
Y

or
X2
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Alternative Model Specifications
Y
True Relationship
X2
25
Alternative Model Specifications

• When there is marked non-linearity in a given
  Y-Xj relationship, it is not appropriate to
  assume a linear relationship between Y-Xj
• Fortunately, some non-linear Y-Xj relations can
  also be estimated using the OLS method (i.e.
  formulas)

26
The Polynomial Specification (7.4)

• A polynomial model specification (with respect to
  only) is

27
The Polynomial Specification
28
The Polynomial Specification

• In a polynomial specification, as Xj increases, Y
  can increase or decrease at an increasing or at
  a decreasing rate it is a very flexible
  non-linear model specification

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The Polynomial Specification
Increasing at a decreasing rate
Y
X2
30
The Polynomial Specification
Increasing at an increasing rate
Y
X2
31
The Polynomial Specification
Decreasing at a decreasing rate
Y
X2
32
The Polynomial Specification
Decreasing at an increasing rate
Y
X2
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The Polynomial Specification

• An advantage of the polynomial model
  specification is that it can combine situations
  in which some of the independent variables are
  non-linearly related to Y while others are
  linearly related to Y
• The reciprocal model specification (to be
  discussed next) also has this advantage

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The Polynomial Specification

• A polynomial model can be estimated by OLS,
  viewing as any other independent variable
  in the multiple regression
• In the example before j1, i.e. a polynomial
  specification with respect to is desired
  both ( and would be included as
  independent variables in the data set given to
  the Excel program for OLS (linear regression)
  estimation

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The Polynomial Specification

• Polynomial models with respect to more than one
  of the independent variables can be similarly
  estimated by OLS

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The Polynomial Specification

• Y513.03 - 1.35X1 0.058X12 24.34X2
• Provide an example in Excel
• Change signs between X1 and X12
• Slope

37
The Reciprocal Specification (6.4)

• The reciprocal model specification is

38
The Reciprocal Specification

• A reciprocal model specification, for example,
  fits Y-Xj relations that look like in the
  previous graph
• As Xj (X1 in the graph) increases, Y increases or
  decreases, but always at a decreasing rate
• In all cases, as Xj gets large Y approaches a
  limit value (which equals 4 in the graphed
  example)

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The Reciprocal Specification

• The slope of a reciprocal model specification
  is
• ,which can be
  positive or negative depending on the sign of
• Unlike the linear model specification, the slope
  is different depending on the value of
  (give numerical example with
  and )

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The Reciprocal Specification

• In other words, the relationship between Y and
  the transformed independent variable
  is linear
• (
  )

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The Reciprocal Specification

• Therefore, the standard OLS method (i.e.
  formulas) can be used to fit this line
• Instead of is used
  as the jth independent variable in the OLS
  formulas or in the data set given to the Excel
  program for calculating the OLS parameter
  estimates

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The Log-Linear Specification (6.5)

• A special type of non-linear relations become
  linear when they are transformed with logarithms
• Specifically, consider
  

43
The Log-Linear Specification
44
The Log-Linear Specification

• Note that is the anti-natural
  logarithm of
  therefore is simply a multiplicative
  constant
• For example, if
  and ( , the model
  is(note that
  )
• In Excel (EXP(4)(101.5))X10.5

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The Log-Linear Specification

• This is also known as the Log-Log or Double-Log
  specification, because it becomes a linear
  relation when taking the natural logarithm of
  both sides
• (notice that and, thus, the
  intercept above is )

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The Log-Linear Specification

• The former implies that the usual OLS formulas
  (or the standard Excel program) can be used to
  estimate the coefficients of a Log-Linear model
  specification, but they are applied to
  and ( ,
  instead of and ,
  ( (i.e. let
  and ( ,
  
  for all is and then use the OLS formulas or the
  Excel program)

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The Log-Linear Specification

• A disadvantage of the log-linear specification is
  that one has to assume that all of the Y-Xj
  relations in the model conform to this type of
  non-linear specification (i.e. one needs to take
  the ln of Y and of all of the independent
  variables in the model)

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The Log-Linear Specification
49
The Log-Linear Specification

• In short, in the Log-Linear model specification
• If as increases
  decreases at a decreasing rate
• If as increases
  increases at a decreasing rate

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The Log-Linear Specification

• If , as increases, Y increases
  at a constant rate (i.e. Y is a linear function
  of ( , but with no intercept)
• If , as increases, Y increases
  at an increasing rate

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The Log-Linear Specification

• Also note that in a Log-Linear specification all
  ( and values must be positive, since
  the natural logarithm of a non-positive number is
  not defined
• An important feature is that directly
  measures the elasticity of Y with respect to Xj
  i.e. the percentage change in Y when Xj changes
  by one percent

52
The Log-Linear Specification

• Notice in this model specification the slope
  (i.e. the unit change in Y when Xj changes by one
  unit) is not constant (it varies for different
  values of Xj), but the elasticity is constant
  throughout!

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The Log-Linear Specification
Elasticity is Constant and Equals 1.75
Increasing Slope
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Assignment

• Using your class project data, estimate a
  multiple regression model using a polynomial
  specification with respect to phosphorous and
  irrigation water use.
• Estimate a model using a polynomial
  specification with respect to phosphorous and a
  reciprocal specification with respect to
  irrigation water use.

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Assignment

• Estimate a multiple regression model using the
  log-linear model specification, and fully
  interpret each of the estimated parameter values

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Assignment

• Graph the following estimated reciprocal models
  for values of from 1 to 10 and
  (

57
Assignment

• Graph the following estimated log-linear models
  for values of from 1 to 10 and (
  

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Assignment

• Interpret the estimated parameter values
  associated to ln(X2i) in each case from 5.