Summary - Multivariate Econometrics (MSc Econometrics and Operations Research)

Summary

Joya da Silva Patricio Gomes

Multivariate Econometrics

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Student Number: 2806884




December 13, 2024

,Contents

Part I: Dynamic regression theory 1
VAR(1) process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Distribution of the VAR(1) process . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Sequence properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Marginalizing, conditioning and exogeneity . . . . . . . . . . . . . . . . . . . . . . 8
The lag operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Autoregressive and moving average dynamic structures . . . . . . . . . . . . . . 12
The simple autoregressive model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Martingale difference processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Properties of the autoregression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Part II: Unit root non-stationarity, Cointegration and Vector Error Correction Models
(VECM) 21
The random walk model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
The probability background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
The unit root autoregression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Spurious regressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Cointegrated time series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Limit theory for cointegrating regressions . . . . . . . . . . . . . . . . . . . . . . . 34
Testing for cointegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
The VECM framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Johansen’s analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Inference in the cointegrating VAR . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Part III: Panel Data Models 49
Cross-sectional dependence in panel data models . . . . . . . . . . . . . . . . . . 49
Nickell bias in short dynamic panel data models . . . . . . . . . . . . . . . . . . . 56




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,Part I: Dynamic regression theory

VAR(1) process
 
xt,1
 xt,2 
 
xt = 
 .. 

 . 
xt,m

• xt is a vector of economic variables (e.g., GDP, interest rates) observed at time t.
• These variables are often interrelated and evolve over time.
{xt : −∞ < t < ∞}
• This sequence is called a ”random sequence” of economic variables.
VAR(1) Model:
E(xt | xt−1 ) = δt + Λxt−1
• δt is a vector of constants (intercepts).
• Λ is an m × m matrix of coefficients showing the influence of past values.
• The model uses the previous time step (xt−1 ) to predict the current value (xt ).


ε t = xt − δt − Λxt−1
• ε t is called the ”mean innovation process” and represents the unpredictable part of
the time series.
Rewritten VAR(1) Model:
xt = δt + Λxt−1 + ε t

Properties of the Error Term (ε t ):
1. The expected value of the error term given past information is zero:

E(ε t | Xt−1 ) = E(xt − δt − Λxt−1 | Xt−1 ) = 0.

2. By the law of iterated expectations, the unconditional expectation of the error term is
zero:
E(ε t ) = 0.

3. The error term is uncorrelated with all lagged values of the variables:

E(ε t x′t− j ) = 0 for all j > 0.

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, Summary Multivariate Econometrics

4. The error term is uncorrelated with its own past values:

E(ε t ε′t− j ) = 0 for all j > 0.

Conditional Distribution of the Error Term:
• To fully specify the data-generating process, we assume a conditional variance for ε t :

E(ε t ε′t | Xt−1 ) = Ω,

where Ω is a constant matrix representing the unconditional variance of ε t .
Gaussian Assumption for Error Term:
• Assume that the conditional distribution of ε t | Xt−1 is Gaussian (normal).


ε t | Xt−1 ∼ MV N (0, Ω),


xt | Xt−1 ∼ MV N (δt + Λxt−1 , Ω),

 
1
Dt (xt | Xt−1 ) = (2π )−m/2 |Ω|−1/2 exp − ε′t Ω−1 ε t .
2
• MV N: Multivariate normal distribution.
• Assuming ε t is Gaussian with fixed mean and variance, and uncorrelated over time
implies that ε t is identically and independently distributed (i.i.d).
Reduced Form Representation:


xt = δt + Λxt−1 + ε t
• This is the reduced form of the model, where all right-hand side variables are prede-
termined at time t.
• No variable directly affects other variables at the same time point (no contemporane-
ous effects) which is generally not in line with economic theory.
Structural Form Representation:


Bxt = Γδt + Cxt−1 + ut ,
• B is a full-rank matrix, representing contemporaneous relationships among the vari-
ables.
• B is defined as:  
1 b12 ... b1m
 .. .. .. 
b
 . . . 

B =  21 ̸= Im
 .. .. .. .. 
 . . . . 

bm1 ... ... 1

• The error term ut has properties:

E(ut | Xt−1 ) = 0, E(ut u′t | Xt−1 ) = Σ.

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