VECM

by Professor Throckmorton
for Time Series Econometrics
W&M ECON 408/PUBP 616
Slides

Summary¶

• Vector Error Correction Model (VECM) is specifically used for analyzing cointegrated time series that are first-order integrated, I(1).

• VECM is builds on the concept of cointegration, i.e., each time series might be I(1), and certain linear combinations of them are stationary, I(0).

• VECM is derived from a reduced-form VAR in levels when the data is cointegrated

• VECM includes an error correction term that reflects the deviation from the lagged long-run average or “equilibrium”

VECM¶

• Suppose $\mathbf{y}_t$ is a VAR($p$) with I(1), possibly cointegrated, variables in levels, then

  $$\begin{gather*} \Delta \mathbf{y}_t = \boldsymbol{\Pi} \mathbf{y}_{t-1} + \sum_{h=1}^{p-1} \boldsymbol{\Gamma}_h \Delta \mathbf{y}_{t-h} + \boldsymbol{\upsilon}_t \end{gather*}$$

  (1)

• $\Delta \mathbf{y}_t$ is the first difference of a vector of $n$ non-stationary, I(1), variables, making them stationary. $\sum_{h=1}^{p-1} \boldsymbol{\Gamma}_h \Delta \mathbf{y}_{t-h}$ captures the short-run dynamics of the system, and $\boldsymbol{\upsilon}_t$ is a vector of white noise errors.

• $\boldsymbol{\Pi} \mathbf{y}_{t-1}$ is the error correction term, i.e., the long-run equilibrium relationship between variables in levels, where $\boldsymbol{\Pi}$ is the cointegration structure.

  • If $\boldsymbol{\Pi} = \boldsymbol{0}$, there is no cointegration and no need for VECM.

  • If $\boldsymbol{\Pi}$ has reduced rank, $r < n$, then $\boldsymbol{\Pi} = \boldsymbol{\alpha} \boldsymbol{\beta}'$, where:

    • $\boldsymbol{\beta}_{n \times r}$ is a matrix of cointegrating vectors.

    • $\boldsymbol{\alpha}_{n \times r}$ is a matrix of adjustment coefficients, determining how each variable returns to its long-run equilibrium.

  • If $\boldsymbol{\Pi}$ is full rank, $r = n$, then $\mathbf{y}_t$ is already stationary and no need for VECM.

Rank of a matrix¶

• A square matrix $\mathbf{A}_{n \times n}$ is full rank if its rank is $n$. It’s not full rank, or has reduced rank, if its rank $<n$.

• The following conditions are all equivalent — if any one is true, the matrix is full rank (warning: this is not a complete list)

  1. Linearly independent columns: No column can be written as a linear combination of the others.

  2. Linearly independent rows: No row is a linear combination of other rows.

  3. Non-zero determinant: $\det(\mathbf{A}) \ne 0$ or $|\mathbf{A}| \ne 0$

  4. Invertibility: there exists $\mathbf{A}^{-1}$ such that $\mathbf{A} \mathbf{A}^{-1} = I_n$

Examples¶

• Consider the following square matrix

  $$\begin{gather*} \mathbf{A} = \begin{bmatrix} 2 & 4 \\ 1 & 2 \end{bmatrix} \end{gather*}$$

  (2)

  • Column 2 is $2\times$ column 1 $\rightarrow$ $\mathbf{A}$ is not full rank

  • Row 1 is $2\times$ row 2 $\rightarrow$ $\mathbf{A}$ is not full rank

• Consider the following square matrix

  $$\begin{gather*} \mathbf{A} = \begin{bmatrix} 3 & 5 \\ 2 & 4 \end{bmatrix} \end{gather*}$$

  (3)

  • We can’t multiply one row or column by a constant to get the other.

  • The determinant of a $2 \times 2$ matrix is $|\mathbf{A}| = ad - bc$

  • Plug in the values $|\mathbf{A}| = (3)(4) - (5)(2) = 12 - 10 = 2$

  • Since $|\mathbf{A}| \neq 0$, the matrix is full rank.

• For a $3 \times 3$ square matrix

  $$\begin{gather*} \mathbf{A} = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \end{gather*}$$

  (4)

• The determinant formula is $a(ei - fh) - b(di - fg) + c(dh - eg)$.

• That is the sum of the three determinants for $2 \times 2$ matrices.

• See Determinant.

Cointegration¶

• If some linear combination of non-stationary I(1) time series is stationary, then the variables are cointegrated, i.e., they share a stable long-run linear relationship despite short-run fluctuations.

• Suppose we want to test for how many cointegrating relationships, $r$, exist among $n$, I(1), time series.

• For example, if $r = 0$, then there are no cointegrating relationships, and if $r=n-1$ then all variables are cointegrated with eachother.

• To justify estimating a VECM, there should be at least one cointegrating relationship between the variables, otherwise just esimate a VAR after first differencing.

Johansen Cointegration Test¶

• First, use the ADF (unit root) test to confirm that all time series are I(1).

• The Johansen Cointegration Test has nested hypotheses

  • Suppose there are $r_0 = 0$ cointegrating relationships

    • Null Hypothesis: $H_0: r \leq r_0$ (i.e., the number of cointegrating relationships $\leq r_0$)

    • Alternative Hypothesis: $H_A: r > r_0$

  • For each null hypothesis $H_0: r \leq r_0$, if the test statistic $>$ critical value $\Rightarrow$ reject $H_0$.

  • Then update $r_0 = 1,2,3,\ldots$ until test fails to reject

• In practice, use coint_johansen() function from statsmodels.tsa.vector_ar.vecm.

  • Choose lag length ($p-1$) for the VAR in differences.

  • Decide on deterministic components (constant/time trend) in the test.

• Proceed to estimate a VECM if cointegration is found.

Example¶

• The U.S. macroeconomy has gone through many important generational changes that might affect the cointegrating relationships between aggregate time series variables

• E.g, The Great Depression, Post-War Period, Great Inflation of the late 1960s to early 1980s, The Great Moderation, The Great Recession (and the slow recovery), Post COVID

• Let’s test whether consumption and income are cointegrated for the 25 years prior to the Great Moderation (1960-1984)

• Measure them with log real PCE and log real GDP

Read Data¶

Create Dataframe¶

Transform Data¶

Plot Sample¶

ADF Unit Root Test¶

• Note: the argument const_trend sets whether the test accounts for a constant, 'c', and time trend, 'ct'.

• Fail to reject null for both time series in levels

• Reject the null for both first-differenced time series

• Since they are both $I(1)$, proceed to Johansen Cointegration Test

Trends¶

• Stochastic trend, i.e., $y_t = \mu + y_{t-1} + \varepsilon_t$

  • Can drift without bound — non-stationary

• Deterministic trend, i.e., $y_t = \mu + \delta t + \varepsilon_t$

  • Deviations from the time trend are mean-reverting

• Both deterministic and stochastic trends, i.e., $y_t = \mu + \delta t + y_{t-1} + \varepsilon_t$

• First differencing will remove either trend.

• In adfuller, setting regression = 'ct' allows for both a constant and linear time trend.

• Fail to reject the null that the data (after removing linear trend) has a unit root, i.e., removing the linear trend did not make the data stationary.

• The data is probably stationary around a stochastic trend.

• Regardless, first differencing the data for the VECM will remove these trends, so we don’t need to account for them later.

Cointegration Test¶

• There is evidence that GDP and consumption are cointegrated.

• det_order = -1 omits a common constant or linear time trend from the cointegrating relationship.

• k_ar_diff = 1 means we are using a VECM with 1 lag of differences.

Estimation¶

• There is evidence that both GDP and consumption are I(1) and they are cointegrated.

• If the data wasn’t cointegrated, they should be first differenced and estimate a VAR instead.

• Proceed with estimating a bivariate VECM

  $$\begin{align*} \Delta \tilde{y}_{t} &= \mu_1 + \alpha_1 (\beta_1 \tilde{y}_{t-1} + \beta_2 \tilde{c}_{t-1}) + \Gamma_{11} \Delta \tilde{y}_{t-1} + \Gamma_{12} \Delta \tilde{c}_{t-1} + \cdots + \varepsilon_{1,t} \\ \Delta \tilde{c}_{t} &= \mu_2 + \alpha_2 (\beta_1 \tilde{y}_{t-1} + \beta_2 \tilde{c}_{t-1}) + \Gamma_{21} \Delta \tilde{y}_{t-1} + \Gamma_{22} \Delta \tilde{c}_{t-1} + \cdots +\varepsilon_{2,t} \end{align*}$$

  (5)

  where $\tilde{y} =$ log real GDP and $\tilde{c} =$ log real PCE and $\beta_1 \tilde{y}_{t-1} + \beta_2 \tilde{c}_{t-1}$ is their cointegrating relationship.

Model Selection¶

In select_order:

• deterministic sets which deterministic terms appear in the VECM, e.g., deterministic = 'co' puts a constant outside the error correction to estimate the intercept since $\Delta \tilde{y}_{t}$ and $\Delta \tilde{c}_{t}$ have non-zero means

• Selection criterion for lag order includes AIC and BIC (among others)

In select_coint_rank:

• method='trace' uses the Johansen Cointegration Test

• det_order=-1 means there are not any deterministic terms in the cointegrating relationship(s), which is the same assumption as above when we used coint_johansen directly (above).

• k_ar_diff is set to result of the AIC selection criterion.

Results¶

Cointegrating Relationship¶

• $\boldsymbol{\Pi} \mathbf{y}_{t-1}$ is the error correction term, i.e., the long-run equilibrium relationship between variables

• If $\boldsymbol{\Pi}$ has reduced rank, $r < n$, then $\boldsymbol{\Pi} = \boldsymbol{\alpha} \boldsymbol{\beta}'$, where $\boldsymbol{\beta}_{n \times r}$ is a matrix of cointegrating vectors.

Error Correction Term¶

• Interpreting the cointegration coefficients, or the error correction term (ECT)

  $$\begin{align*} ECT_{t-1} &= \hat{\beta}' [\tilde{y}_{t-1},\tilde{c}_{t-1}] = \tilde{y}_{t-1} - 0.94 \tilde{c}_{t-1} \\ \rightarrow \tilde{y}_{t-1} &= 0.94 \tilde{c}_{t-1} + ECT_{t-1} \end{align*}$$

  (6)

• The long-run linear relationship between GDP and consumption is strongly positive, $\tilde{y}_t \approx 0.94 \tilde{c}_t$, and the ECT measures how far away the system is from equilibrium.

• $\hat{\alpha}_1 = -0.24$ (and is significant) measures how much GDP corrects disequilibrium, i.e., returns to equilibrium

  • The significance indicates which variables adjust, e.g., $\hat{\alpha}_2$ was not significant so consumption probably does not correct disquilibrium.

  • The sign indicates the direction of adjustment of each variable towards the long-run equilibrium, e.g., since $\hat{\alpha}_1 < 0$, GDP pushes the system back to equilibrium

  • The magnitude is the speed or strength of adjustment, e.g., since $|\hat{\alpha}_1| > |\hat{\alpha}_2|$, GDP adjusts more quickly than consumption.

Result: reject that the ECT has a unit root and conclude that it is probably stationary, as desired.

Impulse Response Functions¶

• The estimated VECM cannot be used to compute IRFs

• IRFs are only defined for a VAR with stationary time series or a VAR with cointegrated time series in levels (e.g., Christiano et al., 2005)

• Instead we would estimate a reduced form VAR($p$) separately to compute IRFs, where $p$ comes from the selected lag order from the VECM estimation

• Then we can ask the question, how much does an increase in income change consumption and for how long?

• Since both GDP and consumption are both clearly endogenous to many other things in the macroeconomy, it doesn’t make sense to use a Cholesky decomposition (e.g., orthogonalization) to estimate shocks.

Forecasting¶

• With the estimated VAR, forecasts can also be made

• Want to compare forecasts to actual data

• This may not be a great example because the 1980s is still relative volatile