Macroeconomic Modelling and Bayesian Methods

Abstract

This paper discusses the evolution of macroeconomic modelling. In particular, it focuses on Bayesian methods and provides some applications of the Bayesian Vector Autoregression methods to the Indian economy.

Appendix A: A Note on Priors

The Minnesota prior (and its variants) used in Bayesian VAR is due to Litterman (1981, 1982, 1986), Doan et al. (1984), Todd (1984) and Spencer (1993). The restrictions on the coefficients specify normal prior distributions with means zero and small standard deviations for all coefficients with decreasing standard deviations on increasing lags, except for the coefficient on the first own lag of a variable that is given a mean of unity.

$$\begin{aligned} \beta _i =N\left( {1,\sigma _{\beta _i }^2 } \right) \;{\text {and}}\; \beta _j =N\left( {0,\sigma _{\beta _j }^2 } \right) \end{aligned}$$

where $\beta _{i}$ denote the coefficients associated with the lagged dependent variables in each equation and $\beta _{j}$ denote all other coefficients.The prior variances $\sigma _{\beta _i }^2$ and $\sigma _{\beta _j }^2$ denote uncertainity about the prior means in each equation.

The standard deviation of the prior distribution for lag m of variable j in equation i for all i, j, and m, $\sigma _{ijm}$, is specified as follows:

$$\begin{aligned} \begin{array}{lll} \sigma _{ijm} &{}=&{} \{wg(m)f(i, j)\}si/sj;\\ f(i, j) &{}=&{} 1, if \;i = j; \\ &{}=&{} k \quad \mathrm{otherwise} \;({ 0< k < 1}); \mathrm{and} \\ g(m)&{}=&{} m^{-d}, d > 0. \end{array} \end{aligned}$$

The term si equals the standard error of a univariate autoregression for variable i. The ratio si / sj scales the variables to account for differences in units of measurement and allows the specification of the prior without consideration of the magnitudes of the variables. The parameter w measures the standard deviation on the first own lag and describes the overall tightness of the prior. The tightness on lag m relative to lag 1 equals the function g(m), assumed to have a harmonic shape with decay factor d. The tightness of variable j relative to variable i in equation i equals the function f(i, j).

The value of f(i, j) determines the importance of variable j relative to variable i in the equation for variable i, higher values implying greater interaction. For instance, $f(i, j) = 0.5$ implies that relative to variable i, variable j has a weight of 50%. Further, this value (k) may differ across j s highlighting the flexibility imparted by the Bayesian approach. A tighter prior occurs by decreasing w, increasing d, and/or decreasing k. Examples of selection of hyperparameters are given in Dua and Ray (1995), Dua and Smyth (1995), Dua and Miller (1996) and Dua et al. (1999), Dua et al. (2003, 2008, 2017), Dua and Ranjan (2012), and Dua and Garg (2017).

The BVAR method uses Thei (1971) mixed estimation technique that supplements data with prior information on the distributions of the coefficients. With each restriction, the number of observations and degrees of freedom artificially increase by one. Thus, the loss of degrees of freedom due to overparameterization does not affect the BVAR model as severely.

Figure 2 provides a graphical representation of the Minnesota prior where all coefficients have zero prior mean, apart from the first own lag. Moreover, it is observed that the prior distributions tend to become more concentrated for coefficients on longer lags. Further, the prior distributions of the lags of the other variables are more concentrated than those of the variable’s own lags.

Fig. 2

Minnesota prior (Source: Blake and Mumtaz 2012, p. 5)

In Fig. 3, we provide an example of a tight and a loose prior. In both the cases, the mean of the normal distribution is 1 but the variance for the tight prior is 2 and that for the loose prior is 4.

Fig. 3

Tight ($y1\sim N\left( {\mu ,\sigma _1 } \right) )$vs. Loose prior ($y2\sim N\left( {\mu ,\sigma _2 } \right) )$ with $\mu =1,\sigma _1 =2,\sigma _2 =4,~and ~\sigma _1 <\sigma _2$