Uncovering time variation in public expenditure multipliers: new evidence

Abstract

The paper provides fresh evidence on the dynamics of public expenditure multipliers and the factors explaining them. The findings of this paper may facilitate the policymakers in effectively framing fiscal policy to expedite economic stabilisation and uplift economic growth. The paper proceeds with computing the time-varying public expenditure multipliers by employing the time-varying parameter–vector autoregressive (TVP–VAR) model on the Indian quarterly data set between 1997Q1 and 2019Q4. It then examines the role of structural factors in explaining the time variation in public expenditure multipliers. The empirical investigation reveals the heterogeneity in time-varying transmission and effectiveness of revenue capital and total expenditure shocks. The study further finds the adverse effects of fiscal instability and trade openness on public expenditure multipliers. The financial development and propensity to consume augment the public expenditure multiplier values. The findings also provide insights into the periodic impact of expenditure multipliers, which is relevant for policymakers.

Appendices

Appendix

1.1 Gibbs sampling procedure

See Table 7.

The TVP–VAR model equations are as follows:

$${X}_{t}={A}_{0,\mathrm{t}}+{A}_{1,t}{X}_{t-1}+{A}_{2,t}{X}_{t-2}+ \dots \dots +{A}_{P,t}{X}_{t-P}+{u}_{t}$$

(5)

$${u}_{t}\sim N(0,{\Sigma }_{\mathrm{t}})$$

$${\Theta }_{t}={\Theta }_{t-1}+{\omega }_{t}$$

(6)

$${\omega }_{t}\sim N(0,\Omega )$$

$$\mathrm{log}{\sigma }_{t}=\mathrm{log}{\sigma }_{t-1 }+ {\varsigma }_{t}$$

(7)

$${\varsigma }_{t}\sim N(0,\Xi )$$

$${\upphi }_{i,t}={\upphi }_{i,t-1}+{\vartheta }_{i,t}$$

(8)

$${\vartheta }_{i,t}\sim N(0,{\Psi }_{\mathrm{i}})$$

The Gibbs procedure for sampling parameter draws is as follows.

Step 1: Initialize ${X}_{t}, {\Theta }^{T}, {\phi }^{T},\Omega ,\Xi ,\Psi , {\mathrm{and }s}^{T}$ based on OLS estimates of linear VAR.

Step 2: The Gibbs sample procedure proceeds by drawing ${\upsigma }_{\mathrm{t}}$ using the Kim et al. (1998) algorithm. The TVP–VAR model in Eq. (1) can be rewritten as

$${F}_{t}^{-1} ({X}_{t}- {A}_{0,t}+{A}_{1,t}{X}_{t-1}+{A}_{2,t}{X}_{t-2}+ \dots \dots +{A}_{P,t}{X}_{t-P})={D}_{t}^{1/2}{e}_{t}$$

$${X}_{t}^{**}={D}_{t}^{1/2}{e}_{t}$$

(9)

Here ${X}_{t}^{**}$ equals ${F}_{t}^{-1}$ (${X}_{t}- {A}_{0,t}+{A}_{1,\mathrm{t}}{X}_{t-1}+{A}_{2,t}{X}_{t-2}+ \dots \dots +{A}_{P,t}{X}_{t-P})$. Given ${\Theta }_{t}$ and ${F}_{t}^{-1}$, ${X}_{t}^{**}$ becomes observable. Equations (9) and (7) form the non-linear state-space model. Hence, we transform it into a linear state-space system by squaring the fifth equation and then taking the log.

$${X}_{t}^{***}=2{h}_{t}+{e}_{t}^{*}$$

(10)

Here ${X}_{t}^{***}$ = $\mathrm{log}{{X}_{t}^{**}}^{2}$ ${h}_{t}=\mathrm{ log}{\sigma }_{t}$ and ${e}_{t}^{*}$ = log (${e}_{t}^{2}$). Equations (7) and (10) form the linear state-space model. However, ${e}_{\mathrm{t}}^{*}$ now follows the log chi-square distribution. Therefore, we approximate the log chi-square distribution by the mixture of seven normal distributions to convert the system into a gaussian linear state-space system. This strategy has been discussed in Kim et al. (1998). The details about the mixture of seven normal distributions with component probabilities ${\mathrm{q}}_{\mathrm{j}}$ mean ${\mathrm{m}}_{\mathrm{j}}$ − 1.2704, and variance ${\mathrm{v}}_{\mathrm{j}}^{2}$ is given in Table 7.

Table 7 Mixture of seven normal distributions

Let ${s}^{T}$= [${s}_{1}$, …, ${s}_{T}$]′ be the vector of indicator variable choosing a distribution from the mixture of seven normal distributions. Given the value of ${s}_{t}=j$, Carter and Kohn, (1994) algorithm used to draw ${\mathrm{h}}_{\mathrm{t}}$ from the distribution of $({e}_{t}^{*}|{s}_{t}=\mathrm{j}$). Here, ${e}_{t}^{*}|{s}_{t}=j\sim N({\mathrm{m}}_{\mathrm{j}}$ − 1.2704, ${v}_{j}^{2}$). More precisely ${h}_{t}$ is drawn from $N({h}_{t|t+1}$,${H}_{t|t+1}$). Here, ${h}_{t|t+1}=E\left( {h}_{t}\right|{h}_{t+1}, {X}_{t}, {\Theta }^{T}, {\upphi }^{T},\Omega ,\Xi ,\Psi , {s}^{T})$ and ${H}_{t|t+1}=\mathrm{VAR}\left( {h}_{t}\right|{h}_{t+1}, {X}_{t}, {\Theta }^{T}, {\upphi }^{T},\Omega ,\Xi ,\Psi , {s}^{T})$ are the conditional mean and variance obtained from the backward recursion equations.

Step 3: In the third step, the Gibbs sampler draws ${\upphi }^{T}$. We can rewrite Eq. 5 as

$${{F}_{t}^{-1}X}_{t}^{*}={D}_{t}^{1/2}{e}_{t}$$

(11)

Here ${X}_{t}^{*}$ equals (${X}_{t}- {A}_{0,t}+{A}_{1,t}{X}_{t-1}+{A}_{2,t}{X}_{t-2}+ \dots \dots +{A}_{P,t}{X}_{t-P})$. Since ${F}_{t}^{-1}$ is a lower triangular matrix with one on the diagonal. The system of equations in A.7 can be written as

$${X}_{i+1,t}^{*}=-{X}_{\left[1,i\right],t}^{*}{\upphi }_{i,t}+{\sigma }_{i+1,t}{e}_{i+1,t} i=2\dots n$$

(12)

Here ${\sigma }_{i,t,} {e}_{i,t}$ are the ith element of ${\sigma }_{t }\mathrm{and }{e}_{t}$. ${X}_{[1,i],t}^{*}$ is the row vector [${X}_{1,t}^{*}$,…… ${X}_{i,t}^{*}$]. Given ${\Theta }^{T}$ and ${\sigma }^{T}$, ${X}_{t}^{*}$ becomes observable, and Eqs. 12 and 8 form the Gaussian linear state-space model, where states are ${\phi }_{i,t}.$ Since ${\phi }_{i,t}$ and ${\phi }_{j,t}$ are independent of each other, the Carter and Kohn, (1994) algorithm applied to draw ${\phi }_{i,t}$ from $N({\upphi }_{i,t|t+1}$,${\Phi }_{i,t|t+1}$). Here, ${\phi }_{i,t|t+1}$ = $E\left( {\phi }_{i,t}\right|{\phi }_{\mathrm{i},t+1}, {X}_{t}, {\Theta }^{T}, {\sigma }^{T},\Omega ,\Xi ,\Psi )$ and ${\Phi }_{i,t|t+1}$= $\mathrm{VAR}\left( {\phi }_{i,t}\right|{\phi }_{i,t+1}, {X}_{t}, {\Theta }^{T}, {\sigma }^{T},\Omega ,\Xi ,\Psi )$.

Step 4: In this step, we draw the regression coefficients ${\Theta }^{T}.$ Given ${F}_{\mathrm{t}}^{-1}$, ${D}_{t}$, $\Omega$, $\Xi$ and ${\Psi }_{\mathrm{i}}$, Eqs. (5) and (6) form the linear state-space model. Kalman filter and backward recursion, as given in Carter and Kohn, (1994), are used to draw ${\Theta }_{t}$. ${\Theta }_{t}$ is drawn from $N({\Theta }_{t|t+1}$,${P}_{t|t+1}$), where ${\Theta }_{t|t+1}=E\left( {\Theta }_{t}\right|{\Theta }_{t+1}, {X}_{t}, {\sigma }^{T}, {\phi }^{T},\Omega ,\Xi ,\Psi )$ and ${P}_{t|t+1}=\mathrm{VAR}\left( {\Theta }_{t}\right|{\Theta }_{t+1}, {X}_{t}, {\sigma }^{T}, {\phi }^{T},\Omega ,\Xi ,\Psi ).$

Step 5: In this step, we draw parameters $\Omega ,\Xi ,\Psi$ from their distributions $p(\Omega$|${X}_{t}, {\Theta }^{T}, {\phi }^{T},{\sigma }^{T})$, $p(\Xi$|${X}_{t}, {\Theta }^{T}, {\phi }^{T},{\sigma }^{T})$, and $p(\Psi$|${X}_{t}, {\Theta }^{T}, {\phi }^{T},{\sigma }^{T})$ respectively. Given ${X}_{t}, {\Theta }^{T}, {\phi }^{T},{\mathrm{and }\sigma }^{T}$ the parameters $\Omega ,\Xi ,\Psi ,$ have Inverse Wishart distribution from which draws are obtained directly (Gelman et al. (1995).

Step 6: This step draws ${s}^{T}$. We sample ${s}_{i,t}$ from discrete density $p({s}_{i,t}=j$|${X}_{i,t}^{***}, {\sigma }_{i,t}$) $\propto {q}_{j}{f}_{N}({X}_{i,t}^{***}|2{\sigma }_{i,t}+{m}_{j}$−1.2704, ${v}_{j}^{2}).$

Step 7: Go to step 2.

1.2 Convergence diagnostics

See Figs. 5, 6, 7, 8, 9, 10, 11, 12, 13, 14.

Fig. 5

Gweek’s, Raftery and Lewis convergence diagnostic for coefficient and volatility parameters in Model 1. Note: X-axis reports the TVP–VAR coefficients, and Y-axis reports their convergence diagnostic value

Fig. 6

Gweek’s, Raftery and Lewis convergence diagnostic for variance of coefficient and volatility parameters in Model 1. Note: X-axis reports the TVP–VAR coefficients, and Y-axis reports their convergence diagnostic value

Fig. 7

Gweek’s, Raftery and Lewis convergence diagnostic for coefficient and volatility parameters in Model 2. Note: X-axis reports the TVP–VAR coefficients, and Y-axis reports their convergence diagnostic value

Fig. 8

Gweek’s, Raftery and Lewis convergence diagnostic for variance of coefficient and volatility parameters in Model 2. Note: X-axis reports the TVP–VAR coefficients, and Y-axis reports their convergence diagnostic value

Fig. 9

Gweek’s, Raftery and Lewis convergence diagnostic for coefficient and volatility parameters in Model 3. Note: X-axis reports the TVP–VAR coefficients, and Y-axis reports their convergence diagnostic value

Fig. 10

Gweek’s, Raftery and Lewis convergence diagnostic for variance of coefficient and volatility parameters in Model 3. Note: X-axis reports the TVP–VAR coefficients, and Y-axis reports their convergence diagnostic value

Fig. 11

Gweek’s, Raftery and Lewis convergence diagnostic for coefficient and volatility parameters in Model 4. Note: X-axis reports the TVP–VAR coefficients, and Y-axis reports their convergence diagnostic value

Fig. 12

Gweek’s, Raftery and Lewis convergence diagnostic for variance of coefficient and volatility parameters in Model 4. Note: X-axis reports the TVP–VAR coefficients, and Y-axis reports their convergence diagnostic value

Fig. 13

Gweek’s, Raftery and Lewis convergence diagnostic for coefficient and volatility parameters in Model 5. Note: X-axis reports the TVP–VAR coefficients, and Y-axis reports their convergence diagnostic value

Fig. 14

Gweek’s, Raftery and Lewis convergence diagnostic for variance of coefficient and volatility parameters in Model 5. Note: X-axis reports the TVP–VAR coefficients, and Y-axis reports their convergence diagnostic value.

1.3 Robustness of regression results

See Tables 8, 9, 10, 11, 12, 13.

Table 8 Bayesian linear regression with dependent variables as PE multipliers at 16 and 20 quarters computed with prior hyperparameters B

Table 9 Bayesian linear regression with dependent variables as PE multipliers at 4, 8, and 12 quarters computed with prior hyperparameters B

Table 10 Bayesian linear regression with dependent variables as PE multipliers at 16 and 20 quarters computed with prior hyperparameters C

Table 11 Bayesian linear regression with dependent variables as PE multipliers at 4, 8, and 12 quarters computed with prior hyperparameters C

Table 12 Bayesian linear regression with dependent variables as PE multipliers at 16 and 20 quarters computed with prior hyperparameters D

Table 13 Bayesian linear regression with dependent variables as PE multipliers at 4, 8, and 12 quarters computed with prior hyperparameters D