Multiple Time Series Model¶

Table of Contents¶

• Intervention Analysis
• Drawbacks of Multivariate Macro Econometric Models
• Introduction to VAR Analysis
• Estimation & Identification
• The Impulse Response Function
• Structural VARs

Some Basics¶

Matrix¶

• Linearly dependent:
  • The matrix determinant is not zero
  • The matrix is not of full rank.

Lets now discuss about eigen values and eigen vectors.

For a square matrix $A$, a non-zero vector $\mathbf{v}$ is called an eigenvector if: $$ A\mathbf{v} = \lambda \mathbf{v} $$ where $\lambda$ is a scalar called the eigenvalue.

A key point to note is that multiplying a scaler implies that only the magnitude of the original vector changes without changing its direction.

To find eigenvalues and eigenvectors of a vector, we need to frame the characteristic equation.

• Characteristic Equation: $(A-\lambda I)V =0$ to find eigen values

$$ A = \begin{bmatrix} 4 & 2 \\ 1 & 3 \end{bmatrix} - \lambda \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} $$

where,

• $A$ is the coefficient matrix of the system of equations (or the vector in question)

For this to have a non-trivial solution, the following condition must be satisfied

$$ \det(A-\lambda I) = 0 $$

The roots of this will give us the $\lambda$ (eigen)values.

Finance¶

• Risk: potential loss of the security under uncertain environment
• Efficient Market Hypothesis
  • Short-run equilibrium
  • Long-run equilibrium
• Capital Asset Pricing Model (CAPM)
  • $R_{p} = R_{f} + \beta(R_{m}) +\epsilon$
  • $\beta$ is the sensitivity in financial terms

In the presence of a structural change, PP test is better and useful to test for stationarity.

Policy Intervention can induce structural breaks or changes in time series.

In order to test for this, we perform Intervention analysis.

Intervention Analysis¶

• Intervention can be single or multiple causing a single/multiple structural(s) in the series.

• Comprises of

  • Dummy Variable
  • Transfer function

Examples of Functions¶

1. Production function types: $Q = AL^\alpha K^\beta$
2. Simplest lienar function

Most econometric equations are linear in nature.

• The Practical difficulty to decie upon which function to use stems from its complexity. Theoretically, a function might make sense for a scenario, but end up being very complex.

• Thus we use simple functions but to model the interventions, add dummy varialbes (for a single intervention) and transfer functions (when the effect of the intervention is continuous/gradual instead of intstantaenous.)

Test for Significance¶

How to check whether intervention is significant or not?

$$ y_{t} = \alpha_{0} + \alpha_{1}y_{t-1} + C_{0}Z_{t} + \epsilon_{t} $$

where, $Z_{t}$ is a dummy (binary) variable

$$ Z_{t} = \begin{cases} 0 & t\lt t_{0} \\ 1 & t \geq t_{0} \end{cases} $$

Suppose, $Z_{t}$ = 1 and $C_{0}$ is constant

To test the significance of $Z_{t}$

$$ \begin{align*} H_{0} &: C_{0} = 0 \\ H_{1} &: C_{0} \neq 0 \end{align*} $$

• $T\text{-test} = \dfrac{C_{0}}{SE(C_{0})}$
• Find critical values
• Compare if $C_{0} \neq 0$ then there is a structural change (i.e. the policy is effective)

Types of Shocks¶

• Pure Jump: After the shock, a jump occurs that persists in a constant manner.

• Impulse Function: Doesn't persist for longer duration

• Gradually Increasing (Decreasing): As the name implies, and the increasing/decreasing nature depends on $C_{0}$.

• Prolonged Impulse: An initial jump whose affect is prolonged but not forever, as the variable gradually returns to its original state as before the intervention.

The Impulse Response Function¶

It refers to a function of lags which shows the effect of a different shock (impulse) on the variables in the equation system, at various leads.

Multiple Policy Shocks¶

Here, our interest is to check the impact of the shock.

• e.g. Trump Tariffs: 10% $\to$ 25% $\to$ 50%
  • Take exports, GDP, Automobile on Agricultural Sector.
  • All the variables are impacted differently and won't reflect in the same way for a certain shock (here, tariffs)

$$ y_{t} = \alpha_{0} +\alpha_{1} y_{t-1} + C_{0}Z_{t} +\epsilon_{t} $$

How to write equations for these multiple shocks?¶

Let $Z_{t}$ be the shock function. Then, for one shock we have,

$$ y_{t} = \alpha_{0} + \alpha_{1} Y_{t-1} + C_{0}Z_{t} + \epsilon_{t} $$

For multiple shocks, $Z_{t}$ appear at multiple time periods ($t = $) using the lag operator ($L$) can also be written as…

$$ \begin{align*} & = \alpha_{0} +\alpha_{1}LY_{t} +C_{0}Z_t + \epsilon_{t} \\ (1- \alpha_{1}L)Y_{t} & = \alpha_{0}+C_{0}Z_{t} + \epsilon_{t} \\ Y_{t} & = \dfrac{1}{(1- \alpha_{1}L)}\left(\alpha_{0}+C_{0}Z_{t} + \epsilon_{t}\right) \\ \end{align*} $$

And thus we get, $$ Y_{t} = \dfrac{\alpha_{0}}{1-\alpha_{1}} + C_{0} \sum_{i=0}^\infty \alpha_{1}^i Z_{t-i} + \sum_{i=0}^\infty \alpha_{1}^i\epsilon_{t} $$

The algebra here is actually simple,

• $\dfrac{1}{1-\alpha_{1}L}$ expands to $1 + (\alpha_{1}L) + (\alpha_{1}L)^{2} +\dots+(\alpha_{1}L)^i+\cdots$ as an infinite geometric series sum
• On the constant $\alpha_{0}$, the lag operator has no effect, so it becomes $\alpha_{0}(1 + \alpha_{1}+\alpha_{1}^{2} +\cdots) = \dfrac{\alpha_{0}}{1-\alpha_{1}}$
• The lag operator series applies to the $Z_{t}$ and $\epsilon_{t}$.

Feature	Transfer Function	Impulse Response Function (IRF)
Form	An operator (a polynomial in the lag operator $L$).	A sequence of real-valued coefficients.
The Math	$$\frac{1}{1-\alpha_1 L}$$	$$\{\alpha_1^i\}_{i=0}^{\infty}$$
Purpose	To define the dynamic relationship between input and output variables.	To show the dynamic path of how the output variable responds to a one-unit shock.
Analogy	The recipe for generating the response.	The numerical value of the response at each point in time.

Impulse response function can also help interpret the impact of one variable, since the variables are interrelated through multiple time lags, creating complex feedback loops.

Hence, we we isolate a "pure" shock to one variable. The error terms in a VAR are often correlated. Thus, to see this pure effect we must first orthogonalize the shocks. The most common method for this is the Cholesky decomposition, which forces a causal order on the variables.

Drawbacks of Multivariate Macro Econometric Models¶

One variable cannot capture the whole economy i.e. one variable cannot be seen in isolation

For for example, to understand one variable of interest we need a set of variables. Say,

$$ \text{Export} \begin{cases} \text{Forex} \\ \cdots \\ \text{BoP} \\ \text{Trade Balance} \end{cases} $$

But there may be many other variables involving around a certain variable.

In a broad sense, economics is the study of the behavior of related economic variables. To answer the question "How each one behaves in accordance to the other".

$$ Y = \beta_{0} + \beta_{1} X_{1} + \beta_{2} X_{2} + \epsilon $$

where,

• $\beta_{i} = \dfrac{\partial{Y}}{\partial{Xi}}$ , the partial derivative… keeping other variables constant. The ceteris paribus condition.

Omitted Variable Bias¶

It requires a lot of variables or information to explain the dependent variable fully. Even if we keep adding variables, we will never be able to exhaust all the necessary variables.

There will still remain some exogenous terms.

• We may forecast or predict that both the series will meet again
• (a) and (b) are in equilibrium in the long- and short-run

1. Even if variables are non-stationary, they can be in equilibrium of some kind. This is called cointegration.
2. If the variables are not in equilibrium, predictions cannot be done. As in figure (II) where both are non-stationary and parallel to each other

Introduction to VAR Analysis¶

While modeling relationships among variables, we encounter situations where the RHS variables are endogenous. These endogenous variables are determined simultaneously by the left hand side variables. Thus, we must formulate a simultaenous model. However, this cannot be done by OLS estimation since in some of these equations, there are variables which are correlated with error terms.

The VAR model, proposed by Christopher Sims, where all variables are treated as endogenous.

Consider multiple one-variable autogregressions.

$$ \underset{\vdots}{V_{1}}\quad \underset{\vdots}{V_{2}}\quad \dots\quad \underset{\vdots}{V_{20}}\ $$

This is a simple extension of AR.

Here, each vector is a time-series variable $V_{1}$,$V_{2}$… so, we are looking at

$$ \begin{pmatrix} y_{11} \\ y_{12} \\ y_{13} \\ \vdots \\ y_{1t} \\ \end{pmatrix} \quad \begin{pmatrix} y_{21} \\ y_{22} \\ y_{23} \\ \vdots \\ y_{2t} \\ \end{pmatrix} \cdots \begin{pmatrix} y_{20,1} \\ y_{20,2} \\ y_{20,3} \\ \vdots \\ y_{20,t} \\ \end{pmatrix} $$

Study of several variables simultaneously $\to$ VAR system for multivariate analysis.

Let's take multiple AR(k) univariate models,

$$ \begin{align} GDP(x_{t}) & = \alpha_{0} +\alpha_{1}x_{t-1} +\alpha_{2}x_{t-2} + \alpha_{k}x_{t-k} \\ \pi(y_{t}) & = \alpha_{0} +\alpha_{1}y_{t-1} +\alpha_{2}y_{t-2} + \alpha_{k}y_{t-k} \\ e(\mathcal{U}_{t}) & = \alpha_{0} +\alpha_{1}\mathcal{U}_{t-1} +\alpha_{2}\mathcal{U}_{t-2} + \alpha_{k}\mathcal{U}_{t-k} \\ \end{align} $$

Such series of equations can be represented in a matrix formLet's take many one-variable autoregressions.

$$ \begin{bmatrix} y_{a\times1} \\ \\ \\ \\ \end{bmatrix}_{m\times 1} = \begin{pmatrix} \\ & A & \\ & \text{Coefficients} & \\ \\ \end{pmatrix}_{m\times (k + 1)} \begin{pmatrix} \\ X \\ \\ \\ \end{pmatrix}_{(k+1)\times1} + \begin{pmatrix} \\ \mathcal{E} \\ \text{Error Matrix} \\ \\ \end{pmatrix}_{m\times1} $$

This representation is considered as vector autoregression. This is the general form. In particular, this is a generalization of AR process.

Contrast it with univariate analysis, where $X$ is a scalar, but here it is a vector.

Specification¶

1. Variables to be included in the ssystem (theoretical background)
2. Inclusion of Deterministic Components (ensure no Omitted variable bias)... include the intercept unless there is reason to believe that all variables have a zero mean.
3. Lag length selection
   • Error vector should not exhibit any autocorrelation. $E(u_{it},u_{js})= 0$ for all $t \neq s$

Uses of VAR¶

1. Forecasting
2. Granger Causality
3. Innovation Accounting
   • Impulse Response Function

Estimation & Identification¶

Nothing yet found here.

Structural VARs¶

Nothing yet found here