Financial Econometrics and Data Science Multivariate Models
Dr Ran Tao
9. Multivariate Models
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9.1 Simultaneous Equations Models
9.2 Tests for Exogeneity
9.3 Indirect Least Squares (ILS)
9.4 Instrumental Variables
9.5 An Example of the Use of 2SLS
9.6 Vector Autoregressive Models (VARs)
9.7 An Example of the Use of VAR
9. Multivariate Models
9.1 Simultaneous Equations Models
9.1 Simultaneous Equations Models
Simultaneous Equations Models
All the models we have looked at thus far have been single equations models of the form
y = Xβ + u
All of the variables contained in the X matrix are assumed to be EXOGENOUS – that is, their values are determined outside the equation.
y is an ENDOGENOUS variable – that is, their values are determined by the equation.
9.1 Simultaneous Equations Models (Cont’d)
An example from economics to illustrate – the demand and supply of new houses:
Qdt = α+βPt+γSt+ut (1) Qst = λ+μPt+κTt+vt (2) Qdt = Qst (3)
Qdt = quantity of new houses demanded at time t
Qst = quantity of new houses supplied (built) at time t Pt = (average) price of new houses prevailing at time t St = price of a substitute (e.g. older houses)
Tt = some variable embodying the state of housebuilding technology, ut and vt are error terms.
9.1 Simultaneous Equations Models
Simultaneous Equations Models: The Structural Form
Assuming that the market always clears, that is, that the market is always in equilibrium, and dropping the time subscripts for simplicity
Q = α+βP+γS+u (4)
Q = λ+μP+κT+v (5)
This is a simultaneous STRUCTURAL FORM of the model.
The point is that price and quantity are determined simultaneously (price affects quantity and quantity affects price).
P and Q are endogenous variables, while S and T are exogenous.
We can obtain REDUCED FORM equations corresponding to (4) and (5) by solving equations (4) and (5) for P and for Q (separately).
9.1 Simultaneous Equations Models
Obtaining the Reduced Form
Solving for Q,
α+βP +γS+u=λ+μP +κT +v (6)
Solving for P,
Q − α − γS − u = Q − λ − κT − v (7) ββββμμμμ
9.1 Simultaneous Equations Models (Cont’d)
Rearranging (6),
βP − μP = λ − α + κT − γS + v − u
(β−μ)P =(λ−α)+κT −γS+(v−u) P=λ−α+ κ T− γ S+v−u (8)
β−μ β−μ β−μ β−μ Multiplying (7) through by βμ,
μQ−μα−μγS−μu=βQ−βλ−βκT −βv
μQ−βQ=μα−βλ−βκT +μγS+μu−βv
(μ−β)Q=(μα−βλ)−βκT +μγS+(μu−βv)
Q=μα−βλ− βκ T+ μγ S+μu−βv (9) μ−β μ−β μ−β μ−β
(8) and (9) are the reduced form equations for P and Q.
9.1 Simultaneous Equations Models
Simultaneous Equations Bias
But what would happen if we had estimated equations (4) and (5), i.e. the structural form equations, separately using OLS?
Both equations depend on P. One of the CLRM assumptions was that E(X′u) = 0, where X is a matrix containing all the variables on the RHS of the equation.
It is clear from (8) that P is related to the errors in (4) and (5) – i.e. it is stochastic.
What would be the consequences for the OLS estimator, βˆ, if we ignore the simultaneity?
Recall that βˆ = (X′X)−1X′y and y = Xβ + u
9.1 Simultaneous Equations Models (Cont’d)
βˆ = βˆ = βˆ =
(X′X)−1X′(Xβ + u) (X′X)−1X′Xβ + (X′X)−1X′u β + (X′X)−1X′u
Taking expectations,
E(βˆ) = E(β) + E((X′X)−1X′u)
E(βˆ) = β + E((X′X)−1X′u)
If cov(ut, xt) = 0, E(X′u) = 0, which would be the case in a single equation system, so that E(βˆ) = β, which is the condition for unbiasedness.
9.1 Simultaneous Equations Models (Cont’d)
But …. if the equation is part of a system, then E(X′u) ̸= 0, in general.
Conclusion: Application of OLS to structural equations which are part of a simultaneous system will lead to biased coefficient estimates.
Is the OLS estimator still consistent, even though it is biased?
No – In fact the estimator is inconsistent as well.
Hence it would not be possible to estimate equations (4) and (5) validly using OLS.
9.1 Simultaneous Equations Models
Avoiding Simultaneous Equations Bias So What Can We Do?
Taking equations (8) and (9), we can rewrite them as
P = π10 + π11T + π12S + ε1 (10) Q=π20 +π21T +π22S+ε2 (11)
We CAN estimate equations (10) & (11) using OLS since all the RHS variables are exogenous.
But … we probably don’t care what the values of the π coefficients are; what we wanted were the original parameters in the structural equations – α, β, γ, λ, μ, κ.
9.1 Simultaneous Equations Models
Identification of Simultaneous Equations
Can We Retrieve the Original Coefficients from the π’s?
Short answer: sometimes.
As well as simultaneity, we sometimes encounter another problem: identification.
Consider the following demand and supply equations
Q = α + βP Supply equation (12)
Q = λ + μP Demand equation (13) We cannot tell which is which!
Both equations are UNIDENTIFIED or NOT IDENTIFIED, or UNDERIDENTIFIED.
9.1 Simultaneous Equations Models (Cont’d)
The problem is that we do not have enough information from the equations to estimate 4 parameters. Notice that we would not have had this problem with equations (4) and (5) since they have different exogenous variables.
9.1 Simultaneous Equations Models
What Determines whether an Equation is Identified or not?
We could have three possible situations: 1. An equation is unidentified
– like (12) or (13)
– we cannot get the structural coefficients from the reduced
form estimates
2. An equation is exactly identified – e.g. (4) or (5)
– can get unique structural form coefficient estimates 3. An equation is over-identified
– Example given later
– More than one set of structural coefficients could be
obtained from the reduced form.
How do we tell if an equation is identified or not? There are two conditions we could look at:
9.1 Simultaneous Equations Models (Cont’d)
– The order condition – is a necessary but not sufficient
condition for an equation to be identified.
– The rank condition – is a necessary and sufficient condition for identification. We specify the structural equations in a matrix form and consider the rank of a coefficient matrix.
Statement of the Order Condition (from Ramanathan 1995, pp.666)
Let G denote the number of structural equations. An equation is just identified if the number of variables excluded from an equation is G-1.
If more than G-1 are absent, it is over-identified. If less than G-1 are absent, it is not identified.
9.1 Simultaneous Equations Models (Cont’d)
In the following system of equations, the Y’s are endogenous, while the X’s are exogenous. Determine whether each equation is over-, under-, or just-identified.
Y1 = α0 +α1Y2 +α3Y3 +α4X1 +α5X2 +u1 (14)
Y2 = β0 +β1Y3 +β2X1 +u2 (15)
Y3 = γ0+γ1Y2+u3 (16)
If # excluded variables = 2, the eqn is just identified If # excluded variables > 2, the eqn is over-identified
9.1 Simultaneous Equations Models (Cont’d)
If # excluded variables < 2, the eqn is not identified Equation 14: Not identified
Equation 15: Just identified
Equation 16: Over-identified
9.2 Tests for Exogeneity
9.2 Tests for Exogeneity
Tests for Exogeneity
How do we tell whether variables really need to be treated as endogenous or not?
Consider again equations (14)-(16). Equation (14) contains Y2 and Y3 but do we really need equations for them?
We can formally test this using a Hausman test, which is calculated as follows:
1. Obtain the reduced form equations corresponding to (14)-(16). The reduced forms turn out to be:
π10 +π11X1 +π12X2 +v1 (17) π20 +π21X1 +v2 (18) π30 +π31X1 +v3 (19)
Estimate the reduced form equations (17)-(19) using OLS,
and obtain the fitted values, Yˆ , Yˆ , Yˆ 123
9.2 Tests for Exogeneity (Cont’d)
2. Run the regression corresponding to equation (14).
3. Run the regression (14) again, but now also including the
fitted values Yˆ21 , Yˆ31 as additional regressors:
Y1 =α0 +α1Y2 +α3Y3 +α4X1 +α5X2 +λ2Yˆ21 +λ3Yˆ31 +u1(20)
4. Use an F-test to test the joint restriction that λ2 = 0, and
λ3 = 0. If the null hypothesis is rejected, Y2 and Y3 should be treated as endogenous.
9.3 Indirect Least Squares (ILS)
9.3 Indirect Least Squares (ILS)
Indirect Least Squares (ILS)
Cannot use OLS on structural equations, but we can validly apply it to the reduced form equations.
If the system is just identified, ILS involves estimating the reduced form equations using OLS, and then using them to substitute back to obtain the structural parameters.
However, ILS is not used much because
1. Solving back to get the structural parameters can be tedious. 2. Most simultaneous equations systems are over-identified.
9.3 Indirect Least Squares (ILS)
Estimation of Systems Using Two-Stage Least Squares
In fact, we can use this technique for just-identified and over-identified systems.
Two stage least squares (2SLS or TSLS) is done in two stages:
Obtain and estimate the reduced form equations using OLS. Save the fitted values for the dependent variables.
Estimate the structural equations, but replace any RHS endogenous variables with their stage 1 fitted values.
9.3 Indirect Least Squares (ILS) (Cont’d)
Example: Say equations (14)-(16) are required. Stage 1:
Estimate the reduced form equations (17)-(19) individually by OLS and obtain the fitted values, .
Replace the RHS endogenous variables with their stage 1 estimated values:
Y1 = α0 +α1Yˆ21 +α3Yˆ31 +α4X1 +α5X2 +u1 (21) Y2 = β0+β1Yˆ31+β2X1+u2 (22) Y3 = γ0+γ1Yˆ21+u3 (23)
9.3 Indirect Least Squares (ILS) (Cont’d)
Now Yˆ21 and Yˆ31 will not be correlated with u1, Yˆ31 will not be correlated with u2, and Yˆ21 will not be correlated with u3.
It is still of concern in the context of simultaneous systems whether the CLRM assumptions are supported by the data.
If the disturbances in the structural equations are autocorrelated, the 2SLS estimator is not even consistent.
The standard error estimates also need to be modified compared with their OLS counterparts, but once this has been done, we can use the usual t- and F-tests to test hypotheses about the structural form coefficients.
9.4 Instrumental Variables
9.4 Instrumental Variables
Instrumental Variables
Recall that the reason we cannot use OLS directly on the structural equations is that the endogenous variables are correlated with the errors.
One solution to this would be not to use Y2 or Y3 , but rather to use some other variables instead.
We want these other variables to be (highly) correlated with Y2 and Y3, but not correlated with the errors - they are called INSTRUMENTS.
9.4 Instrumental Variables (Cont’d)
Say we found suitable instruments for Y2 and Y3, z2 and z3 respectively. We do not use the instruments directly, but run regressions of the form
Y2 = λ1 + λ2z2 + ε1 (24) Y3 = λ3 + λ4z3 + ε2 (25)
Obtain the fitted values from (24) & (25), Yˆ21 and Yˆ31, and replace Y2 and Y3 with these in the structural equation.
We do not use the instruments directly in the structural equation.
It is typical to use more than one instrument per endogenous variable.
9.4 Instrumental Variables (Cont’d)
If the instruments are the variables in the reduced form equations, then IV is equivalent to 2SLS, so that the latter can be viewed as a special case of the former.
What Happens if We Use IV / 2SLS Unnecessarily?
The coefficient estimates will still be consistent, but will be inefficient compared to those that just used OLS directly.
The Problem With IV
What are the instruments? Solution: 2SLS is easier.
Other Estimation Techniques
1. 3SLS - allows for non-zero covariances between the error terms.
9.4 Instrumental Variables (Cont’d)
2. LIML - estimating reduced form equations by maximum likelihood
3. FIML - estimating all the equations simultaneously using maximum likelihood
9.5 An Example of the Use of 2SLS
9.5 An Example of the Use of 2SLS
Who Captures the Power of the Pen?
You, Zhang, and Zhang (2018) Introduction
– How government control affects the roles of the media as an information intermediary and a corporate monitor?
– How state-controlled and market-oriented media differs in China?
Why two types of mass media are different? Consider 4 possibilities:
1. Market-oriented media might write favourable articles for their advertising clients.
2. Market-oriented media might produce sensational news stories to cater for their readers.
9.5 An Example of the Use of 2SLS (Cont’d)
3. State-owned media has less incentive and reward for the efforts in identifying newsworthy stories and high-quality articles.
4. Self-censorship also undermines the quality of news articles by the state-controlled media, particularly when getting involved in political interests.
Informative Market Hypothesis
It predicts that the market-oriented media should have more informative coverage and take a more effective role in monitoring managers.
9.5 An Example of the Use of 2SLS
The sample of media coverage is from 2004 to 2014 based on the eight largest nation-wide business newspaper in China.
China Securities Journal, Securities Daily, Securities Times, Shanghai Securities Journal, China Business Journal, First Financial Daily, The Economic Observer, and 21st Century Business Herald.
The Economic Observer and China Business Journal are issued weekly, others are issued daily.
The eight newspaper’ headquarters are located in four cities: Beijing, Shanghai, Guangzhou, and Shenzhen.
The state-owned newspapers are China Securities Journal, Securities Daily, Securities Times, and Shanghai Securities Journal.
9.5 An Example of the Use of 2SLS (Cont’d)
The market-oriented newspapers are China Business Journal, First Financial Daily, The Economic observer, and 21st Century Business Herald.
Article tone is classified as positive, neutral, or negative using a dictionary method.
Two separated variables are created: MktTone and GovTone, respectively.
Forced top executive turnover, denoted as CEOTurnover, is constructed for the reason of dismissal and legal disputes as an indicator of corporate governance.
9.5 An Example of the Use of 2SLS
Corporate Governance Role of the Media
CEOTurnoveri,t+1 = α0 + β1MktTonei,t(GovTonei,t) + β2Xi,t + ei
where Xi,t includes media coverage, the return-on-assets ratio, annual stock return, the log of total assets, financial leverage, block ownership, state-owned enterprise, board size, the board size of independent directors, CEO age, CEO tenure, and CEO/Chairman duality, firm and year fixed effects.
9.5 An Example of the Use of 2SLS
9.5 An Example of the Use of 2SLS
Interpret Results
In Models (1) and (2), the coefficient of MktTone is significantly negative at the 1% level, while the coefficient on GovTone is not statistically significant.
As shown in Models (4) and (5), the interaction term ROA and MktTone is significantly positive, while the interaction term between ROA and GovTone is not significant.
Model (6) pools all variables and reports similar results.
Poorly performing firms are more likely to have their top executive removed only if they are covered negatively by the market-oriented media.
Negative coverage by the state-controlled media does not have the same effects.
9.5 An Example of the Use of 2SLS
An Instrumental Variable Approach
MktTone and GovTone could jointly determined by other unobservable firm characteristics.
The direction of causality may run from top executive turnover to media coverage rather than vice versa.
If top executive turnover events are predictable, media outlets may follow these events solely to cater to the needs of their audience.
Two instrumental variables are used.
The hometown of newspaper editors.
Individuals who originally come from the same place are more likely to share common culture and social norms. These editors may be more affirmative about their hometown firms.
9.5 An Example of the Use of 2SLS (Cont’d)
Individual who originate from the same place are more likely to establish social connections as they may know the executives of hometown firms. The newspapers are less likely to cast negative reports about these firms.
The introduction of a high-speed rail between the cities of a firm’s and a newspaper’s headquarters.
Journalists can easily conduct an onsite visit with high-speed trains. Newspapers are more likely to be creditable.
“Economic distance" between newspapers and firms is shortened. Newspapers may have an incentive to write more positive articles to keep and develop business firms that can be reached via high-speed rail.
9.5 An Example of the Use of 2SLS
9.5 An Example of the Use of 2SLS
9.6 Vector Autoregressive Models (VARs)
9.6 Vector Autoregressive Models (VARs)
Vector Autoregressive Models
A natural generalisation of autoregressive models popularised by Sims
A VAR is in a sense a systems regression model i.e. there is more than one dependent variable.
Simplest case is a bivariate VAR
y1t = β10 +β11y1t−1 +···+β1ky1t−k +α11y2t−1 +···
+α1ky2t−k + u1t
y2t = β20 +β21y2t−1 +···+β2ky2t−k +α21y1t−1 +··· +α2ky1t−k + u2t
9.6 Vector Autoregressive Models (VARs) (Cont’d)
where uit is a white noise disturbance term with E(uit) = 0, (i = 1, 2), E(u1tu2t) = 0.
The analysis could be extended to a VAR(g) model, or so that there are g variables and g equations.
9.6 Vector Autoregressive Models (VARs)
Vector Autoregressive Models: Notation and Concepts
One important feature of VARs is the compactness with which we can write the notation. For example, consider the case from above where k=1.
We can write this as
y1t = β10 + β11y1t−1 + α11y2t−1 + u1t
y2t = β20 + β21y2t−1 + α21y1t−1 + u2t
y1t β10 β11 α11y1t−1 u1t
y=β+αβy+u 2t 20 21 21 2t−1 2t
9.6 Vector Autoregressive Models (VARs) (Cont’d)
or even more compactly as
yt = β0 + β1yt−1 + ut
g × 1 g × 1 g × gg × 1 g × 1
This model can be extended to the case where there are k
lags of each variable in each equation:
yt = β0 + β1yt−1 + β2yt−2 +···+ βkyt−k + ut g×1 g×1 g×gg×1 g×gg×1 g×gg×1 g×1
We can also extend this to the case where the model includes first difference terms and cointegrating relationships (a VECM).
9.6 Vector Autoregressive Models (VARs)
Vector Autoregressive Models Compared with Structural Equations Models
Advantages of VAR Modelling
– Do not need to specify which variables are endogenous or
exogenous - all are endogenous
– Allows the value of a variable to depend on more than just its own lags or combinations of white noise terms, so more general than ARMA modelling
– Provided that there are no contemporaneous terms on the right hand side of the equations, can simply use OLS separately on each equation
– Forecasts are often better than “traditional structural” models. Problems with VAR’s
– VAR’s are a-theoretical (as are ARMA models) – How do you decide the appropriate lag length?
9.6 Vector Autoregressive Models (VARs) (Cont’d)
– So many parameters! If we have g equations for g variables and we have k lags of each of the variables in each equation, we have to estimate (g + kg2) parameters. e.g. g=3, k=3, parameters=30
– Do we need to ensure all components of the VAR are stationary?
– How do we interpret the coefficients?
9.6 Vector Autoregressive Models (VARs)
Does the VAR Include Contemporaneous Terms?
So far, we have assumed the VAR is of the form y1t = β10 + β11y1t−1 + α11y2t−1 + u1t
y2t = β20 + β21y2t−1 + α21y1t−1 + u2t
But what if the equations had a contemporaneous feedback
y1t = β10 + β11y1t−1 + α11y2t−1 + α12y2t + u1t y2t = β20 + β21y2t−1 + α21y1t−1 + α22y1t + u2t
9.6 Vector Autoregressive Models (VARs)