Economic uncertainty, precautionary motive and the augmented form of money demand function

Abstract

A notable feature of empirical research on the inclusion of the economic uncertainty in the money demand function is that very few published papers rely on the specifications with formal clarity by studying the response of precautionary demand for money to changes in economic uncertainty. To overcome this shortcoming, this study uses the augmented form to specify the demand for money function and examines the empirical validity based on a sample of eleven countries, including four developed countries and seven developing countries. Using the panel error-correction technique, the findings provide some policy implications; the augmented form of money demand function can characterize the difference between the three primary motives of money demand—the precautionary motive, the transactions motive and the speculative motive, and support the choice of narrow money as a guide for monetary policy that ensures that economic agents have enough money to meet unexpected expenses triggered by economic uncertainty and helps to fine-tune the interest rate misalignment and output disruption that eventually improves the effectiveness of monetary policy.

Appendix

Solution Method. We list here the equations describing the grid search algorithm to determine the optimal form of economic uncertainty function that can be used to calculate the index of economic uncertainty.¹⁸ Following the grid search method, as described in Eq. (27), one must minimize the central bank’s loss function subject to the structural macroeconomic model (i.e., the system of equations); the structural macroeconomic model encompasses the IS curve, as described in the first relation, the Phillips curve, as described in the second relation, the reduced form of the exchange rate, as described in the third relation, the contemporaneous economic uncertainty function, as described in the fourth relation, and the monetary policy reaction function, as described in the fifth relation (Note that the form of the fourth relation is non-optimal, but it can be useful for shaping the optimal form of economic uncertainty function via the grid search algorithm). The loss function contains three stabilization objectives, namely the inflation, the output and the interest rate, which is expected to seek to minimize the central bank’s discretionary decision making.

$$ \left\{ {\begin{array}{*{20}l} {\text{Minimizing the central bank's loss function}} \hfill \\ {L = \mu_{{\pi_{g} }} V_{{\pi_{g} }} + \mu_{{y_{g} }} V_{{y_{g} }} + \gamma_{{r_{g} }} V_{{r_{g} }} } \hfill \\ {\text{subject to the structural macroeconomic model}} \hfill \\ {y_{{g_{t} }} = \alpha_{1} y_{{g_{t - 1} }} - \lambda_{1} r_{{g_{t - 1} }} - \delta_{1} e_{{g_{t - 1} }} + \varepsilon_{t} } \hfill \\ {\pi_{{g_{t} }} = \alpha_{2} y_{{g_{t - 1} }} + \beta_{{\pi_{1} }} \pi_{{g_{t - 1} }} - \delta_{2} e_{{g_{t - 1} }} + \eta_{t} } \hfill \\ {e_{{g_{t} }} = \lambda_{2} r_{{g_{t} }} + \upsilon_{t} } \hfill \\ {eu_{t} = \alpha_{3} y_{{g_{t} }} + \beta_{{\pi_{2} }} \pi_{{g_{t} }} - \delta_{3} e_{{g_{t} }} - \lambda_{3} r_{{g_{t} }} + \varpi_{t} } \hfill \\ {r_{{g_{t} }} = \alpha_{4} y_{{g_{t - 1} }} + \beta_{{\pi_{3} }} \pi_{{g_{t - 1} }} - \delta_{4} e_{{g_{t - 1} }} + eu_{t} + \zeta_{t} } \hfill \\ \end{array} .} \right. $$

(27)

Notes: $\mu_{{\pi_{g} }} ,$$\mu_{{y_{g} }} ,$ and $\gamma_{{r_{g} }}$ are the weights attached to the stabilization of the inflation gap, the real output gap, and the real interest rate gap, respectively; $V_{{\pi_{g} }} ,$$V_{{y_{g} }}$ and $V_{{r_{g} }}$ are the unconditional variance of the inflation gap, the real output gap, and the real interest rate gap, respectively; $y_{{g_{t} }} ,$$\pi_{{g_{t} }} ,$$e_{{g_{t} }} ,$$eu_{t}$ and $r_{{g_{t} }}$ are the real output gap, the inflation gap, the real exchange rate gap, the economic uncertainty index, and the real interest rate gap, respectively; considering the system of equations, the theoretical signs for each parameter are $\propto_{1} > 0,$$\lambda_{1} < 0,$$\delta_{1} < 0,$$\propto_{2} > 0,$$\beta_{{\pi_{1} }} > 0,$$\delta_{2} < 0,$$\lambda_{2} > 0,$$\propto_{3} > 0,$$\beta_{{\pi_{2} }} > 0,$$\delta_{3} < 0,$$\lambda_{3} < 0,$$\propto_{4} > 0,$$\beta_{{\pi_{3} }} > 0$ and $\delta_{4} < 0;$$\varepsilon_{t} ,$$\eta_{t} ,$$\upsilon_{t} ,$$\varpi_{t}$ and $\zeta_{t}$ represent the shocks of demand, supply, the exchange rate, economic uncertainty, and the monetary policy, respectively.

To apply the grid search algorithm, one can specify the system of equations of Eq. (27) in the matrix system as:

$$ D_{1} X_{t} = D_{2} X_{t - 1} + Q_{t} . $$

(28)

Thereby,

$$ \left[ {\begin{array}{*{20}l} 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & { - \lambda_{2} } \hfill & 0 \hfill \\ { - \alpha_{3} } \hfill & { - \beta_{{\pi_{2} }} } \hfill & {\delta_{3} } \hfill & 1 \hfill & {\lambda_{3} } \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & { - 1} \hfill & 1 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & { - 1} \hfill & 1 \hfill \\ \end{array} } \right]\;\left[ {\begin{array}{*{20}l} {y_{{g_{t} }} } \hfill \\ {\pi_{{g_{t} }} } \hfill \\ {e_{{g_{t} }} } \hfill \\ {eu_{t} } \hfill \\ {r_{{g_{t} }} } \hfill \\ {\Delta r_{{g_{t} }} } \hfill \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} {\alpha_{1} } \hfill & 0 \hfill & { - \delta_{1} } \hfill & 0 \hfill & { - \lambda_{1} } \hfill & 0 \hfill \\ {\alpha_{2} } \hfill & {\beta_{{\pi_{1} }} } \hfill & { - \delta_{2} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ {\alpha_{4} } \hfill & {\beta_{{\pi_{3} }} } \hfill & { - \delta_{4} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & { - 1} \hfill & 0 \hfill \\ \end{array} } \right]\;\left[ {\begin{array}{*{20}l} {y_{{g_{t - 1} }} } \hfill \\ {\pi_{{g_{t - 1} }} } \hfill \\ {e_{{g_{t - 1} }} } \hfill \\ {eu_{t - 1} } \hfill \\ {r_{{g_{t - 1} }} } \hfill \\ {\Delta r_{{g_{t - 1} }} } \hfill \\ \end{array} } \right] + \left[ {\begin{array}{*{20}l} {\varepsilon_{t} } \hfill \\ {\eta_{t} } \hfill \\ {\nu_{t} } \hfill \\ {\varpi_{t} } \hfill \\ {\zeta_{t} } \hfill \\ 0 \hfill \\ \end{array} } \right], $$

where $Q_{t}$ is the vector of serially uncorrelated disturbances with mean zero, and the covariance matrix associated with $Q$ is given by ${{\varOmega }} = E\left( {QQ^{\prime}} \right)$; to capture the real interest rate gap volatility, an identity formula, i.e., ${{\Delta }}r_{{g_{t} }} = r_{{g_{t} }} - r_{{g_{t - 1} }} ,$ is included in the system. Then, the above form can be equivalently written as X_t = BX_t − 1 + W_t, but with $B \,{\equiv}\, D_{1}^{ - 1} D_{2}$, $W \equiv D_{1}^{ - 1} Q_{t}$ and $~\Sigma = E\left[ {WW^{\prime } } \right] = E\left[ {\left( {D_{1}^{{ - 1}} Q} \right)\left( {D_{1}^{{ - 1}} Q_{t} } \right)^{'} } \right] = D_{1}^{{ - 1}} \Omega \left( {D_{1}^{{ - 1}} } \right)^{\prime } ;$${{\varSigma }}$ is the covariance matrix associated with $W$. Bearing in mind that estimating the system of Eq. (28) estimates B, W, and Q.

Next, the optimization procedure consists in solving:

$$ \left\{ {\begin{array}{*{20}c} {\left\{ {\mathop {\alpha_{3}^{\text{optimal}} , \beta_{{\pi_{2} }}^{\text{optimal}} , \delta_{3}^{\text{optimal}} , \lambda_{3}^{\text{optimal}} }\limits^{\text{Min}} } \right\}} & {L_{t} = \mu_{{\pi_{g} }} V_{{\pi_{g} }} + \mu_{{y_{g} }} V_{{y_{g} }} + \gamma_{{r_{g} }} V_{{r_{g} }} } \\ {s.t.}& {X_{t} = BX_{t - 1} + W_{t} } \\ \end{array} } \right.. $$

In this procedure, the optimal form of the economic uncertainty index, as described in Eq. (29), can be determined, which consists of the optimal combination of the parameters—$\alpha_{3}^{\text{optimal}} ,$$\beta_{{\pi_{2} }}^{\text{optimal}} ,$$\delta_{3}^{\text{optimal}}$ and $\lambda_{3}^{\text{optimal}}$—of the endogenous economic variables that minimizes $L$. The optimal combination of the parameters is globally optimal because the grid search approach gives a global solution.

$$ eu_{t}^{\text{optimal}} = \alpha_{3}^{\text{optimal}} y_{{g_{t} }} + \beta_{{\pi_{2} }}^{\text{optimal}} \pi_{{g_{t} }} - \delta_{3}^{\text{optimal}} e_{{g_{t} }} - \lambda_{3}^{\text{optimal}} r_{{g_{t} }} . $$

(29)

Following Ball (1997) and Svensson (2000), the vector form of the unconditional contemporaneous covariance matrix of X, noted $\upsilon ,$ can be given by,

$$ vec \left( \upsilon \right) = \left[ {I - B \otimes B} \right]^{ - 1} vec \left( \varSigma \right). $$

(30)

The unconditional variance of the inflation gap, the real output gap, and the real interest rate gap, i.e., $V_{{\pi_{g} }} ,$$V_{{y_{g} }} ,$ and $V_{{r_{g} }} ,$ are obtained, respectively, by choosing the appropriate elements in $vec \left( \upsilon \right).$ In this process, $V_{{\pi_{g} }}$ is the first element of $vec \left( \upsilon \right),$$V_{{y_{g} }}$ is the eighth element of $vec \left( \upsilon \right),$ and $V_{{r_{g} }}$ is the thirty-sixth element of $vec \left( \upsilon \right).$ One can define the lowest value of the loss function ($L$) as the sum of the first, eighth, and thirty-sixth elements of $vec \left( \upsilon \right),$ respectively, weighted by $\mu_{{\pi_{g} }} = 1.0,$$\mu_{{y_{g} }} = 1.0,$ and $\gamma_{{r_{g} }} = 0.25.$¹⁹ Precisely, given a combination ($\alpha_{3} ,$$\beta_{{\pi_{2} }} ,$$\delta_{3} ,$$\lambda_{3}$) in solving this sequence, the approach can be symbolized by the sequence below:

$$ (\alpha_{3} ,\beta_{{\pi_{2} }} ,\delta_{3} ,\lambda_{3} ) \Rightarrow B, W \; and \; \varSigma \Rightarrow vec\, \upsilon \Rightarrow L. $$

(31)

There are several computer programming software products that could be used to execute the grid search algorithm to determine the optimal form of economic uncertainty function. The grid search algorithm calibrates the above structural model using the Regression Analysis of Time Series (RATS) program. Table 4 gives results for an optimal form of economic uncertainty function.²⁰ Because of space limitations, this section does not present a description of variables in gap form (i.e., $y_{{g_{t} }} ,$$\pi_{{g_{t} }} ,$$e_{{g_{t} }}$ and $r_{{g_{t} }}$) used in the evaluation of the optimal form of economic uncertainty function, and an estimation method to obtain the estimates of the parameters for solving the calibration problem of identifying an optimal form of economic uncertainty function; however, these presentations are available in Section A and Section B of the supplemental material (Note that the supplemental material can be downloaded at http://alpha.upsi.edu.my/nextcloud/index.php/s/6qFiXAc94DzbZrA).

Table 4 Optimal parameters, unconditional variances of goal variables, losses (results depend on $\mu_{{\pi_{g} }} ,$$\mu_{{y_{g} }} ,$$\gamma_{{r_{g} }}$) and optimal form of economic uncertainty function