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Geen diatitel

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LPNC [log (price of new car)] LPUC [log (price of used car)] LPPB [log (price of public transport) ... Old and New Models. Example. Dummy Variable ... – PowerPoint PPT presentation

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Title: Geen diatitel


1
Time Series Analysis Section 2
2
Time Series Analysis
  • Section 2
  • Autocorrelation (continue)
  • Moving Average and Autoregressive Processes
  • Analytical Example

3
Again, Cant we use Regression?
  • Yes we can still use Least-Squares (LS)
    Regression
  • BUT
  • Cautions are
  • Results from regression are coefficients (ß) and
    standard error (? t-stat)
  • Coefficients are still the same
  • BUT variance (? standard error) is not the same
  • Conclusion
  • Least Squares Method produces unbiased parameter
    estimates but biased standard errors

Autocorrelation
4
Stochastic Processes
  • We will consider two stochastic processes namely
  • Moving Average Process (MA)
  • Autoregressive Process (AR)

Autocorrelation
5
Moving Average Process
  • Suppose that Zt is a purely random process with
    mean zero and variance sz2
  • Then a process Xt is said to be a moving
    average process of order q MA(q) if
  • Xt ß0Zt ß1Zt-1 ßqZt-q

Autocorrelation
6
MA Processes
  • MA processes have been used in many areas,
    particularly econometrics.
  • For example, economic indicators are affected by
    a variety of random events such as strikes,
    government decisions, shortages of key materials
    and so on.
  • Such indicators to a lesser extent in several
    subsequent periods, as so it is at least
    plausible that an MA process may be appropriate.

Autocorrelation
7
Autoregressive Processes
  • Suppose that Zt is a purely random process with
    mean zero and variance sz2
  • Then a process Xt is said to be an
    autoregressive process of order p if
  • Xt a1Xt-1 apXt-p Zt
  • This is rather like a multiple regression model
    but Xt is regressed not on independent variables
    but on past values of Xt.
  • The abbreviation is AR(p) process

Autocorrelation
8
AR processes
  • AR processes have been applied to many situations
    in which it is reasonable to assume that the
    present value of time series depends on the
    immediate past values together with a random
    error.
  • For example, AR(1)
  • Yt ß0 ß1X1 ß2X2 a1Yt-1

Lag Variable
Autocorrelation
9
Example of AR(1)
Autocorrelation
10
Example of AR(1)
Autocorrelation
11
Tests for AR(1) Error Term
  • Consider the following sample error term
  • et Yt Yt Yt (abXt)
  • Durbin-Watson Test
  • We can use the sample residuals to come up with a
    sample estimate of the AR(1) parameter

Autocorrelation
12
Durbin-Watson Test
  • Under following Nulls
  • H0 ? 0, Ha ? gt 0, Test for Positive AR (1)
    parameter
  • H0 ? 0, Ha ? lt 0, Test for Negative AR (1)
    parameter

Autocorrelation
13
Example
Example
14
Variables
Example
15
Gas vs. Year
Example
16
Price of Gas vs. Year
Example
17
Correlation Matrix for Variables
Very high correlation
  • Correlation coefficients gt 0.8 there is a
    multicolinearity problem
  • Keep this in mind for further analysis

Example
18
Demand Function
  • The log-linear model is
  • In logs,
  • This equation is also known as the constant
    elasticity form as in this equation, the
    elasticity of y with respect to changes in x is

Example
19
First Model
  • Check elasticities

Example
20
Omitted Variables
  • Can use the procedures that described in
    Goldberger (1998) p. 94-95
  • Or, in an easier way, looking at correlation
    coefficients and t-stat
  • Either cases lead to drop
  • LPNC log (price of new car)
  • LPUC log (price of used car)
  • LPPB log (price of public transport)

Example
21
Old and New Models
Example
22
Dummy Variable
  • From Graph, we can see two distinct trends
    before 1973 and after 1973
  • DV 1 for the period between 1960 and 1973.
  • DV 0 for the period between 1974 and 1995.
  • We might expect that the different both intercept
    and slope terms in both log (income) and log (p
    of gas)
  • So the model is as follows

Example
23
Model with Dummy Variable
Example
24
New (Final) Model
Example
25
Autocorrelation
  • First check Durbin-Watson Statistic
  • H0 ? 0 if d lt d
  • where d is the appropriate critical value for
    the distribution of under and a test for
    negative autocorrelation can be based on 4-d
  • The Durbin-Watson statistic for our model is
    0.764. The 5 percent critical values for 36
    observation and 5 regressors from the tables are
    1.18 (dL) and 1.80 (dU). Therefore, we may reject
    the null hypothesis of no autocorrelation.
  • And the t-value on the lag error term is 4.312
    with the probability value of 0.0002. Hence we
    may reject the null hypothesis that the AR(1)
    parameter is zero.

Example
26
AR(1) Corrected Results
Example
27
Questions
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