Econometric Analysis with NonLinear Models

1
Econometric Analysis with Non-Linear Models

• By Artak Yengoyan
• (Ph.D, M. Sc. in Economics and Management, M.A.,
  B.A.)
• AFP Armenia
• Contact information
• E-Mail scartyen_at_yahoo.com
• Tel. [phone]

2
Contents

• Introduction
• Description of SETAR models
• Estimation of US GDP with linear models
• Application of the SETAR models to the US GDP
  data
• Estimation of EU GDP with linear models
• Estimation of EU GDP using SETAR models
• Forecasting with SETAR Models
• Conclusions

3
Introduction

• Although a lot of papers are devoted to the
  analysis and explanation of business cycle
  phenomena, it is still one of the complex and
  puzzled aspects of the economy.
• Different ways of explanation of the business
  cycle and different methodology of analysis are
  proposed.
• Potter (1995) tried to detect the important
  aspects of the US GDPs time series properties,
  using the nonlinear time series models instead of
  linear models.

4
Introduction

• It is interesting that nonlinear models for
  post-1945 US GDP data bring to the conclusion
  that although the economy was affected by the
  negative shocks similar to Great Depression
  output returns quick to its trend.
• Linear models applied to the same time period
  show that there is no stability and output
  remains below its trend for a long period.

5
Description of SETAR models

• Self-exciting Threshold Autoregressive Models
  became to interest many economists due to Tong
  (1983, 1990) contributions in this field of
  research.
• One of the aspects that makes nonlinear models
  more attractive in the sense of being a better
  forecasting tool, is the possibility of asymmetry
  generation, which seems to be a characteristic
  feature of macroeconomic variables like GDP,
  employment etc.

6
Description of SETAR models

• The application of the SETAR models by different
  researchers in estimating different
  macroeconomics variables have shown that SETAR
  models have advantages over the univariate linear
  models.
• Rothman (1998) has explored the US unemployment
  rate, Pippenger and Goering (1998) the exchange
  rate data using the SETAR models and in their
  papers have shown that the model provides better
  forecast opportunity compared to the linear
  models.

7
Description of SETAR models

• The self-exciting threshold autoregressive
  (SETAR) model assumes that a variable yt is a
  linear autoregression within a regime, but may
  move between regimes depending on the value taken
  by a lag of yt say, yt-d, so that d is the
  length of the delay.
• Hence, the model is linear within a regime, but
  liable to move between regimes as the process
  crosses a threshold.

8
Description of SETAR models

• Potter (1995) shows how the Self-exciting
  Threshold Autoregressive Model, as some other
  nonlinear models, can be derived as a special
  case of the so called Single Index Generalized
  Multivariate Autoregressive (SIGMA) model.
• For that reason assume Yt is our observation
  variable, which in economic analysis usually
  denote the first difference of the US GDP, Zt
  an unobserved variable, additionally, let Ht
  denote the single index from the history of Yt,
  Zt, F() is a function to the unit interval.

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Description of SETAR models

• So the univariate first order Single Index
  Generalized Multivariate Autoregressive (SIGMA)
  model will be.
• Yt a1 a2F(Ht) ß1 ß2F(Ht)Yt-1 ?1
  ?2F(Ht)Vt
• Where Vt is IDD (0,1)

10
Description of SETAR models

• Potter (1995) derives following special cases of
  the SIGMA model
• If a2 ß2 ?2 0, the we have AR(1) model.
• If F(Ht) Zt and Zt is a two stage Markov
  Chain, then we have the regime switching model of
  Hamilton.
• If F(Ht) 1(Yt-d gt r) then we have a SETAR (d,
  r) model, where 1(A) is an indicator function
  equal to 1 of event A occurs and zero otherwise,
  d is known as a delay parameter and r is the
  threshold parameter.
• If F(Yt-1 - r)/d is a cumulative distribution
  function, then we get Smooth Transition
  Autoregression (STAR) model.

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Description of SETAR models

• So the SETAR model can be written as following.
• Yt v1 ? a1i Yt-i(1 I(Yt-dr)) v2 ?
  a2i Yt-iI(Yt-dr) ut
• with ut IID(0,s2).

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Description of SETAR models

• Most of researchers use two regime SETAR models
  with different delay parameters for analysis of
  economic data.
• Tiao and Tsay (1991) applied four regimes model
  (Regime 1, if Yt-2 lt 0 and Yt-2 gt Yt-1 - a
  worsening recession, Regime 2, if Yt-2 lt 0 and
  Yt-2 lt Yt-1 - a improving recession, Regime 3, if
  Yt-2gt 0 and Yt-2 gt Yt-1 - an expansion with
  negative growth, Regime 4, if Yt-2gt 0 and Yt-2lt
  Yt-1 - an expansion with increasing growth).

13
Estimation of US GDP with linear models

• To show the difference between the linear and
  nonlinear, in particular, SETAR models estimation
  results, usually researchers apply both models to
  a certain data.
• I will try to analyze the US and EU GDPs with
  linear autoregressive and SETAR models to detect
  whether the nonlinear SETAR models provide better
  estimation tools for the economic data analysis.

14
Description of SETAR models

• Potter (1995) analyses the post-Second World War
  quarterly seasonally adjusted US GDP data for
  1947Q1-1990Q4 time period.
• For the analysis autoregressive model of order 5
  was selected.

15
Estimation of US GDP with linear models Linear
model results 1948Q3-1990Q4
16
Estimation of US GDP with linear models

• I would like to start with the linear
  autoregressive models, to find the appropriate
  model which provides the best estimation results
  for the US GDP data 1947Q1-2003Q3.
• For the selection of the model I use the Akaike
  Information Criterion (AIC). The highest order of
  the autoregressive models I consider is pmax 8.
  A model which generates the minimum AIC will be
  selected as the best fit linear model.

17
Estimation of US GDP with linear models Linear
model results 1948Q3-1990Q4
18
Estimation of US GDP with linear models

• As we can see, the lowest value of the AIC is in
  the case of AR(1) model, then the second lowest
  is the AR(4) model.
• However, in case of AR(4) model, the residuals
  Q-statistics has shown that the null hypothesis,
  that the residuals coefficients are zero, can
  not be rejected both at 5 and 10 significance
  levels,
• This can not be said about AR(1) model.
• So I will use AR(4) models results to compare
  them to the SETAR models results.

19
Estimation of US GDP with linear models
20
Application of the SETAR models to the US GDP data

• Lets us switch to the SETAR models application.
• For SETAR model we need to define the delay
  parameter and the threshold value.
• Potter (1995) used the values of d2 and r0.

21
Application of the SETAR models to the US GDP data

• Applying these delay and threshold parameters two
  ways of estimation are available.
• First approach is to split the data and run
  separate regression models. In this case,
  different parameters and variances for each
  regression will be available. This is the method
  which I apply for my estimations.
• The other approach is to run a general model
  without splitting the data and get the same
  variance in both regimes.

22
Application of the SETAR models to the US GDP data

• The application of SETAR model with 5 lags, r0
  and delay d2 for the estimation of US GDP
  1948Q3-1990Q4 by Potter (1995) has given the
  following results.

23
Application of the SETAR models to the US GDP
data SETAR model without restrictions
1948Q3-1990Q4
24
Application of the SETAR models to the US GDP data

• Regime 1 is the constractionary regime and regime
  2 is expansionary regime.
• It should be mentioned that testing results have
  shown that AR3 and AR4 coefficients are not
  significantly different from zero at the 5
  level.
• For that reason, the restricted model was
  estimated by Potter (1995) setting AR3 and AR4
  coefficients equal to zero.

25
Application of the SETAR models to the US GDP data

• Positive results were obtained, in particular,
  the AIC was less than for the unrestricted model
  and no autocorrelation of residuals were
  detected.
• It is important to mention that AR5 lag helps to
  improve the fit of the model and as it comes
  later out the resulted model without lag 5 will
  be worse fit also analyzing data from 1947Q1 to
  2003Q3.
• Another interesting aspect is the negative value
  of AR(2) coefficient in the contractionary regime
  (Regime 1, Yt-2lt 0). Note that having (Yt-2lt 0)
  and negative coefficient in front of it, we will
  get positive effect of the growth.

26
Application of the SETAR models to the US GDP
data SETAR model with restrictions 1948Q3-1990Q4
27
Application of the SETAR models to the US GDP data

• I have estimated the SETAR models for the US GDP
  1947Q1-2003Q3 using the original 5lags model
  without restrictions and SETAR model with
  restricted AR3 and AR4 coefficient equal to zero.
  
• The results show that the restricted model
  provides the better fit compared to the
  unrestricted model. Heteroskedasticity test shows
  that, both at 5 and 10 significance levels, the
  null hypothesis can not be rejected for both
  unrestricted and restricted models.
• Linearity test is rejected in both cases for 1
  significance level.

28
Application of the SETAR models to the US GDP
data SETAR model without restrictions
1947Q1-2003Q3
29
Application of the SETAR models to the US GDP
data SETAR model without restrictions
1947Q1-2003Q3
30
Application of the SETAR models to the US GDP
data SETAR model without restrictions
1947Q1-2003Q3
31
Application of the SETAR models to the US GDP
data SETAR model without restrictions
1947Q1-2003Q3
32
Application of the SETAR models to the US GDP
data SETAR model without restrictions
1947Q1-2003Q3
33
Application of the SETAR models to the US GDP
data SETAR model with restrictions 1947Q1-2003Q3
34
Application of the SETAR models to the US GDP
data SETAR model with restrictions 1947Q1-2003Q3
35
Application of the SETAR models to the US GDP
data SETAR model with restrictions 1947Q1-2003Q3
36
Application of the SETAR models to the US GDP
data SETAR model with restrictions 1947Q1-2003Q3
37
Application of the SETAR models to the US GDP
data SETAR model with restrictions 1947Q1-2003Q3
38
Estimation of EU GDP with linear models

• To analyze the quarterly aggregate GDP data of
  the fifteen countries that compose the European
  Union, for time period of 1960Q1-1994Q4,
  Crespo-Cuaresma (2000) has applied the linear
  autoregressive models of different orders.
• As a criteria of selection of the best linear
  autoregressive model Crespo-Cuaresma (2000) uses
  Akaike Information Criteria.

39
Estimation of EU GDP with linear modelsLinear
AR model estimation results 1960Q1-1994Q4
40
Estimation of EU GDP with linear models

• My analysis of the EU data for 1960Q1-2002Q1 has
  shown that using the same AIC criterion as the
  order selection parameter AR(3) model, will
  provide the best estimation.
• AR(3) model has Akaike Information Criteria
  -7.379209 and sum of squared residuals 0.005744.
• Q-statistics has shown that the null hypothesis,
  that the residuals coefficients are zero, can
  not be rejected both at 5 and 10 significance
  levels.

41
Estimation of EU GDP with linear modelsLinear
AR model estimation results 1960Q1-2002Q1
42
Estimation of EU GDP with linear modelsLinear
AR model estimation results 1960Q1-2002Q1
43
Estimation of EU GDP using SETAR models

• Crespo-Cuaresma (2000) has applied a two-regime
  SETAR model with 3 lags for the estimation of the
  European GDP data for the same time period
  1960Q1-1994Q4. The delay parameter was chosen the
  one which provides the least sum of squared
  residuals compared to the SETAR models with
  different delay parameters.
• As a result, the delay parameter was chosen to be
  one. The application of that model has shown the
  threshold value approximately equal to zero.

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Estimation of EU GDP using SETAR models
Choosing appropriate SETAR model
45
Estimation of EU GDP using SETAR models

• I have applied a two-regime SETAR model with up
  to five lags for the EU GDP data of period
  1960Q1-2002Q1.
• My estimations have shown that the AIC criteria
  for the SETAR model with 3 lags and delay
  parameter equal to 1 is -7.5378. But the
  residuals perform the evidence of some
  autocorrelation slightly above the insignificance
  level.
• A little bit better results, in terms of the
  model fit and residuals, I have got applying 4
  lags model with the same delay parameter, which
  has AIC criteria -7.5450. The residuals
  correlogram has a better performance,
  autocorrelations and partial autocorrelations are
  within insignificance level in 4 lags model.

46
Estimation of EU GDP using SETAR models

• Heteroskedasticity test shows that, both at 5 and
  10 significance levels, the null hypothesis can
  not be rejected for both models.
• The linearity test is rejected in both cases for
  1 significance level. In both cases threshold
  was equal to zero.
• For the delay parameter equals to 2, 3lags SETAR
  model generates AIC criteria equal to -7.4537 and
  the threshold 0.0026490.

47
Estimation of EU GDP using SETAR modelsSETAR
model of EU GDP 1961Q1 2002Q1
48
Estimation of EU GDP using SETAR models

• As we can see the coefficient of the delay
  parameter, that is AR1, is positive in the regime
  of contraction (Regime 1, Yt-1lt 0) as we have
  seen in the case of US GDP, where the coefficient
  of the second lag in the regime of contraction
  (Regime 1, Yt-2lt 0) was negative.
• In other words, there is no observable positive
  effect on the growth in case of EU GDP as it was
  in case of US GDP.

49
Estimation of EU GDP using SETAR modelsSETAR
model of EU GDP 1961Q1 2002Q1
50
Estimation of EU GDP using SETAR modelsSETAR
model of EU GDP 1961Q1 2002Q1
51
Estimation of EU GDP using SETAR modelsSETAR
model of EU GDP 1961Q1 2002Q1
52
Estimation of EU GDP using SETAR modelsSETAR
model of EU GDP 1961Q1 2002Q1
53
Estimation of EU GDP using SETAR modelsSETAR
model of EU GDP 1961Q1 2002Q1
54
Estimation of EU GDP using SETAR modelsSETAR
model of EU GDP 1961Q1 2002Q1
55
Forecasting with SETAR Models

• In this section I would like to present two
  methods of forecasting using the SETAR models.
• The problems that can arise in the process of
  forecasting using nonlinear models and, in
  particular, SETAR models are that if () is a
  nonlinear function, E() ? (E).

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Forecasting with SETAR Models

• The first method, which is applied by the
  researchers, is the Monte Carlo (MC) method.
• Using the SETAR model of the EU GDP described
  above with delay parameter equal to 1 and data up
  to period T forecasting ?T1 is easy as in this
  case the regime of the SETAR process is known as
  yT is known, and the one-step forecast for the MC
  procedure is just
• ?T1 (a0 a1yT apyT-p1) I(yTlt?) (ß0
  ß1yT ßpyT-p1) I(yTgt?).

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Forecasting with SETAR Models

• However, when forecasting ?T2 we only have ?T1,
  which differs from actual yT1 by an error term
  e, in order to decide on the regime in which the
  process is.
• For a given realization of the error process,
  Jesus Crespo-Cuaresma (2000) makes the forecasts
  for period T 2 following way
• ?jTh (a0 a1?Th-1 ap?T-ph)
  I(?Th-1lt?)
• (ß0 ß1?Th-1 ßp?T-ph) I(?Th-1gt?) ej
  Th.
• Simulating J replications of the error process,
  the forecast for ?Th would be
• ?Th 1/J ? ?j Th

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Forecasting with SETAR Models

• The second method which is called naive or
  skeleton (SK) method, uses the approximation
  that E() (E).
• Basically, this method can be interpreted as a
  special case of the MC method in which the errors
  are set to zero.

59
Forecasting with SETAR Models

• Once the forecasts were obtained, Jesus
  Crespo-Cuaresma (2000) computed the following
  statistics for each step, model and procedure,
• Root Mean Squared Error (RMSE) 1/N?(At -
  Ft)21/2
• Mean Absolute Deviation (MAD) 1/N?At - Ft,
• Theil's U statistic (U) RMSE/1/N?At2
• Confusion Rate (CR) Number of wrongly forecasted
  moves (up/down)/Number of observations to be
  forecasted.
• At and Ft are the actual and forecasted values
  respectively.

60
Forecasting with SETAR Models

• On the basis of these statistics it is possible
  to compare different models, in particular,
  linear and non linear SETAR models as forecasting
  tools.
• Jesus Crespo-Cuaresma (2000) has compared the
  forecasts of a SETAR model on European GDP with
  those of a simple AR model by means of a Monte
  Carlo and simple naive methods and shown that
  self-exciting threshold models really perform
  better, than AR models.

61
Conclusions

• Linear models of estimation of business cycles,
  which were mainly used by macroeconomists since
  World War II, partly because they were the only
  techniques and partly because the economic
  theories usually tested in a linear form.
• Linear models, in fact, can hide some interesting
  aspects, which can be detected by the use of
  nonlinear models.
• My estimations of the US and EU GDP data has
  shown that, indeed, the performance of the SETAR
  models are better in comparison to the linear AR
  models and they provide better fit and can be
  used as a better forecasting tool.