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Econometric ModelingEconometric Modeling
Research MethodsResearch Methods
Professor Lawrence W. LanProfessor Lawrence W. LanEmail:Email: [email protected]@mdu.edu.tw
http://140.116.6.5/mdu/http://140.116.6.5/mdu/
Institute of ManagementInstitute of Management
mailto:[email protected]:[email protected]://140.116.6.5/mdu/http://140.116.6.5/mdu/http://140.116.6.5/mdu/mailto:[email protected]8/3/2019 Eco No Metric ModelING
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OutlineOutline
Overview
Single-equation Regression Models
Simultaneous-equation RegressionModels
Time-Series Models
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OverviewOverview
Objectives
Model building
Types of models
Criteria of a good model
Data
Desirable properties of estimators
Methods of estimation
Software packages and books
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ObjectivesObjectives
Empirical verification of the theories in business,economics, management and related disciplines isbecoming increasingly quantitative.
Econometrics, oreconomic measurement, is a socialscience in which the tools of economic theory,mathematical statistics are applied to the analysis ofeconomic phenomena.
Focus on models that can be expressed in equation formand relating variables quantitatively.
Data are used to estimate the parameters of theequations, and the theoretical relationships are testedstatistically.
Used forpolicy analysis and forecasting.
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Model BuildingModel Building
Model building is a science and art, which
serves for policy analysis and forecasting.
science: consists of a set of quantitative tools
used to construct and test mathematical
representations of the real world problems.
art: consists of intuitive judgments that occur
during the modeling process. No clear-cutrules for making these judgments.
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Types of Models (1/4)Types of Models (1/4)
Time-series models
Examine the past behavior of a time series in
order to infer something about its future
behavior, without knowing about the causalrelationships that affect the variable we are
trying to forecast.
Deterministic models (e.g. linearextrapolation) vs. stochastic models (e.g.
ARIMA, SARIMA).
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Types of Models (2/4)Types of Models (2/4)
Single-equation models
With causal relationships (based on
underlying theory) in which the variable (Y)
under study is explained by a single function(linear or nonlinear) of a number of variables
(Xs)
Y: explained or dependent variable Xs: explanatory or independent variables
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Types of Models (3/4)Types of Models (3/4)
Simultaneous-equation models (or multi-
equation simulation models)
With causal relationships (based on
underlying theory) in which the dependent
variables (Ys) under study are related to each
other as well as to a set of equations (linear or
nonlinear) with a number of explanatoryvariables (Xs)
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Types of Models (4/4)Types of Models (4/4)
Combination of time-series and regression
models
Single-input vs. multiple-input transfer function
models
Linear vs. rational transfer functions
Simultaneous-equation transfer functions
Transfer functions with interventions oroutliers
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Criteria of a Good ModelCriteria of a Good Model
Parsimony
Identifiability
Goodness of fit Theoretical consistency
Predictive power
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DataData
Sample data: the set of observations from themeasurement of variables, which may comefrom any number of sources and in a variety offorms.
Time-series data: describe the movement of anyvariable over time.
Cross-section data: describe the activities of anyindividual or group at a given point in time.
Pooled data: a combination of time-series andcross-section data, also known as panel data,longitudinal or micropanel data.
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Desirable Properties of EstimatorsDesirable Properties of Estimators
Unbiased: the mean or expected value of anestimator is equal to the true value.
Efficient (best): the variance of an estimator is
smaller than any other ones. Minimum mean square error(MSE): to trade offbias and variance. MSE is equal to the square ofthe bias and the variance of the estimator.
Consistent: the probability limit of an estimatorgets close to the true value. It is a large-sampleor asymptotic property.
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Methods of EstimationMethods of Estimation
Ordinary least squares (OLS)
Maximum likelihood (ML)
Weighted least squares (WLS)
Generalized least squares (GLS)
Instrumental variable (IV)
Two-stage least squares (2SLS)
Indirect least squares (ILS)
Three-stage least squares (3SLS)
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Software Packages and BooksSoftware Packages and Books
LIMDEP: single-equation and
simultaneous-equation regression models
SCA: time series models
Textbooks (1) Damodar Gujarati, Essentials of Econometrics, 2nd
ed. McGraw-Hill, 1999.
(2) Robert S. Pindyck and Daniel L. Rubinfeld,Econometric Models and Economic Forecasts, 4th ed.
McGraw-Hill, 1997.
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Single-equation Regression ModelsSingle-equation Regression Models
Assumptions
Best Linear Unbiased Estimation (BLUE)
Hypothesis testing Violations for assumptions 1 ~ 5
Forecasting
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AssumptionsAssumptions
A1: (i) The relationship between Y and X is trulyexistent and correctly specified. (ii) Xs arenonstochastic variables whose values are fixed.(iii) Xs are not linearly correlated.
A2: The error term has zero expected value forall observations.
A3: The error term has constant variance for all
observations A4: The error terms are statistically independent.
A5: The error term is normally distributed.
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Best Linear Unbiased EstimationBest Linear Unbiased Estimation
Gauss-Markov (GM) Theorem: Given
assumptions 1, 2, 3, and 4, the estimation of the
regression parameters by least squares (OLS)
method are the best (most efficient) linearunbiased estimators. (BLUE)
GM theorem applies only to linear estimators
where the estimators can be written as a
weighted average of the individual observationson Y.
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Hypothesis TestingHypothesis Testing
Normal, Chi-square, t, and F distributions
Goodness of fit
Testing the regression coefficients (singleequation)
Testing the regression equation (joint
equations) Testing for structural stability or
transferability of regression models
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A1(i) Violation -- Specification ErrorA1(i) Violation -- Specification Error
Omitting irrelevant variables biased and
inconsistent estimators
Inclusion of irrelevant variables
unbiased but inefficient estimators
Incorrect functional form (nonlinearities,
structural changes) biased and
inconsistent estimators
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A1(ii) Violation Xs Correlated with ErrorA1(ii) Violation Xs Correlated with Error
OLS leads to biased and inconsistent estimators
Criteria of good instrumental (proxy) variables
Instrumental-variables estimation consistent,
but no guarantee for unbiased or uniqueestimators
Two-stage least squares (2SLS) estimation
optimal instrumental variable, unique consistentestimators
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A1(iii) Violation -- MulticollinearityA1(iii) Violation -- Multicollinearity
Perfect collinearity between any of Xsno solution will exist
Near or imperfect multicollinearity large
standard error of OLS estimators or widerconfidence intervals; high R2 but fewsignificant t values; wrong signs forregression coefficients; difficulty inexplaining or assessing the individualcontribution of Xs to Y.
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Detection of MulticollinearityDetection of Multicollinearity
Testing the significance of R-i2 from the various
auxiliary regressions. F=[R-i2/(k-1)]/[(1-R-i
2)/(n-k)],
where n=number of observations, k=number of
explanatory variables including the intercept.Check if F-value is significantly different from zero. If yes
(F-value > F-table), X-i and Xi are significantly collinear
with each other.
Variance inflation factor (VIF = 1/(1-R-i2): VIF=1representing no collinearity; if VIF>10 then high degree of
multicollinearity
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A2 Violation Measurement Error in YA2 Violation Measurement Error in Y
OLS will result in biased intercept;
however, the estimated slope parameters
are still unbiased and consistent.
Correction for the dependent variable
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A3 Violation -- HeteroscedasticityA3 Violation -- Heteroscedasticity
It happens mostly for cross-sectional data;
sometimes for time-series data.
OLS will lead to inefficient estimation, but still
unbiased. Can be corrected by weighted least squares
(WLS) method
Detection: Goldfeld-Quandt test, Breusch-Pagantest, White test, Park-Glejser test, Bartlett test,
Peak test, Spearmans rank correlation test, etc.
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A4 Violation -- AutocorrelationA4 Violation -- Autocorrelation
It happens mostly for time-series data;sometimes for cross-sectional data.
OLS will lead to inefficient estimation, but still
unbiased. Can be corrected by generalized least squares(GLS) method
Detection: Durbin-Watson test, runs test. (For
lagged dependent variable, DW2 even whenserial correlation, do not use DW test, use h testor t test instead)
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A5 Violation Non-normalityA5 Violation Non-normality
Chi-square, t, F tests are not valid;however, these tests are still valid for largesample.
Detection: Shapiro-Wilk test, Anderson-Darling test, Jarque-Bera (JB) test.JB=(n/6)[S2 + (K-3)2/4] where n=samplesize, K=kurtosis, S=skewness. (Fornormal, K=3, S=0) JB~ Chi-squaredistribution with 2 d.f.
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ForecastingForecasting
Ex post vs. ex ante forecast
Unconditional forecasting
Conditional forecasting
Evaluation of ex post forecast errors
means: root-mean-square error, root-mean-square
percent error, mean error, mean percent error, mean
absolute error, mean absolute percent error, Theils
inequality coefficient
variances: Akaike information criterion (AIC), Schwarz
information criterion (SIC)
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Simultaneous-equationsSimultaneous-equations
Regression ModelsRegression Models
Simultaneous-equation models
Seemly unrelated equation models
Identification problem
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Simultaneous-equations ModelsSimultaneous-equations Models
Endogenous variables exist on both sides of theequations
Structural model vs. reduced form model
OLS will lead to biased and inconsistentestimation; indirect least squares (ILS) methodcan be used to obtain consistent estimation
Three-stage least squares (3SLS) method will
result in consistent estimation 3SLS often performs better than 2SLS in terms
of estimation efficiency
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Seemly Unrelated Equation ModelsSeemly Unrelated Equation Models
Endogenous variables appear only on the
left hand side of equations
OLS usually results in unbiased but
inefficient estimation
Generalized least squares (GLS) method
is used to improve the efficiency Zellner
method
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Identification ProblemIdentification Problem
Unidentified vs. identified (over identified
and exactly identified)
Order condition
Rank condition
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Time-series ModelsTime-series Models
Time-series data
Univariate time series models
Box-Jenkins modeling approach Transfer function models
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Time-series DataTime-series Data
Yt: A sequence of data observed at equally
spaced time interval
Stationary vs. non-stationary time series
Homogeneous vs. non-homogeneous time
series
Seasonal vs. non-seasonal time series
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Univariate Time Series ModelsUnivariate Time Series Models
Types of models: white noise model,autoregressive (AR) models, moving-average(MA) models, autoregressive-moving average(ARMA) models, integrated autoregressive-
moving average (ARIMA) models, seasonalARIMA models
Model identification: MA(q) sampleautocorrelation function (ACF) cuts off; AR(p)sample partial autocorrelation function (PACF)cuts off; ARMA(p,q) both ACF and PACF dieout
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Box-Jenkins Modeling ApproachBox-Jenkins Modeling Approach
Tentative model identification (p, q) extended
sample autocorrelation function (EACF)
Estimation (maximum likelihood estimation
conditional or exact) Diagnostic checking (t, R2, Q tests, sample ACF
of residuals, residual plots, outlier analysis)
Application (using minimum mean squared errorforecasts)
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Transfer Function ModelsTransfer Function Models
Single input (X) vs. multiple input (Xs) models Linear transfer function (LTF) vs. rational
transfer function (RTF) models Model identification (variables to be used; b, s, r
for each input variable using corner tablemethod; ARMA model for the noise)
Model estimation: maximum likelihoodestimation (conditional or exact)
Diagnostic checking: cross correlation function(CCF)
Forecasting: simultaneous forecasting
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Simultaneous Transfer FunctionSimultaneous Transfer Function
(STF) Models(STF) Models
Purposes (to facilitate forecasting andstructural analysis of a system, and toimprove forecast accuracy)
Yt and Xt can be both endogenousvariables in the system
Use LTF method for model identification,
FIML for estimation, CCM (crosscorrelation matrices) for diagnosticchecking, simultaneous forecasting
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Transfer Function Models withTransfer Function Models with
Interventions or OutliersInterventions or Outliers
Additive Outlier (AO)
Level Shift (LS)
Temporary Change (TC) Innovational Outlier (IO)
Intervention models