A meta-analysis of the multiplier effects of the money supply on prices

Abstract

This paper seeks to identify what drives the heterogeneity of the empirical results regarding the macro-e conomic effect of the money supply on prices. We carried out a meta-regression analysis on a sample of 261 estimates of the quantity theory of money’s coefficient and obtained a set of characteristics possibly impacting the estimated values. On statistical (and other) grounds, we found it appropriate to investigate two separate subsamples: a subsample consisting of multiplier effects obtained from dynamic linear models with or without error correcting mechanisms or from VECM models, and the other subsample consisting of the multiplier effects obtained from the static linear models or cointegration relation or models with static simultaneous equations. Once the potential biases linked to the authors’ publication strategies had been taken into account, the literature led to two very distinct types of effects: an effect in which money has little or no impact on prices, and an effect in which money plays on prices in accordance with what macroeconomic theory suggests (with a multiplier effect close to 1). On the other hand, the nature of the econometric model used to estimate these multiplier effects appeared to play a major role in the explanation for this heterogeneity of the published results.

Appendices

Appendices

1.1 Appendix 1: Funnel plots distinguishing the effects across different specificities of studies

See Figs. 5, 6, 7, 8, 9, 10 and 11.

Fig. 5

Funnel plot: Multiplier effects obtained from different money supply indicator

Fig. 6

Funnel plot: Multiplier effects obtained from different data frequency

Fig. 7

Funnel plot: Multiplier effects obtained from different data transformation

Fig. 8

Funnel plot: Multiplier effects obtained from different estimation technique

Fig. 9

Funnel plot: Multiplier effects obtained from different economic zone

Fig. 10

Funnel plot: Multiplier effects obtained from different countries

Fig. 11

Funnel plot: Multiplier effects across time

1.2 Appendix 2: How to calculate long-term effect in a dynamic model?

Supposing we have the following dynamic model:

$$\begin{aligned} p_{t} = \underset{(\sigma _{1})}{a_{1}} p_{t-l} + \underset{(\sigma _{2})}{a_{2}} m_{t} + \underset{(\sigma _{3})}{a_{3}} m_{t-k} + \epsilon \end{aligned}$$

$a_{1}, a_{2}, a_{3}$ are the estimated coefficients and $\sigma _{1}, \sigma _{2}, \sigma _{3}$ are the standard deviations corresponded. Based on the assumption that $t=t-1=\cdots =t-k=t-l$ in the long-run, the equation above could be rewritten in the following alternative form:

$$\begin{aligned} (1-a_{1}) p = (a_{2} + a_{3})m \text { or } p = \frac{(a_{2} + a_{3})}{(1-a_{1})}m \end{aligned}$$

By calculating the first derivative of p as a function of m, we obtain the coefficient representing the long-term multiplier effect money-price:

$$\begin{aligned} MUL = \frac{\Delta p}{\Delta m} = \frac{a_{2}+a_{3}}{1-a_{1}} \end{aligned}$$

Then, in order to understand how MUL changes with changes in $a_{1}, a_{2}, a_{3}$, we continue to calculate the first derivative of MUL as a function of $a_{1}, a_{2}, a_{3}$:

$$\begin{aligned} \Delta MUL= & {} \frac{1}{1-a{1}} \Delta a_{2} + \frac{1}{1-a{1}} \Delta a_{3} + \frac{a_{2}+a_{3}}{(1-a_{1})^2}\Delta a_{1} \\= & {} \frac{1}{1-a{1}} (\Delta a_{2} + \Delta a_{3}) + \frac{a_{2}+a_{3}}{(1-a_{1})^2}\Delta a_{1} \end{aligned}$$

And so on we deduce the corresponding standard error (Table 8):

$$\begin{aligned} \sigma (MUL) = \left| \frac{1}{1-a_{1}}\right| (\sigma _{2}+\sigma _{3}) + \left| \frac{a_{2}+a_{3}}{(1-a_{1})^2}\right| (\sigma _{1}) \end{aligned}$$

Table 8 Some examples of dynamic models

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