Reconsidering the welfare cost of inflation in the US: a nonparametric estimation of the nonlinear long-run money-demand equation using projection pursuit regressions

Abstract

This paper, first, estimates the appropriate, log–log or semi-log, linear long-run money-demand relationship capturing the behavior US money demand over the period of 1980:Q1–2010:Q4, using the standard linear cointegration procedures found in the literature, and the corresponding nonparametric version of the same based on projection pursuit regression (PPR) methods. We then, compare the resulting welfare costs of inflation obtained from the linear and nonlinear money-demand cointegrating equations. We make the following observations: (i) the appropriate money-demand relationship for the period of 1980:Q1–2010:Q4 is captured by a semi-log function; (ii) based on the estimation of semi-log cointegrating equations, the welfare cost of inflation was found to at the most lie between 0.0131 % of GDP and 0.2186 % of GDP for inflation rates between 0 and 10 %, and; (iii) in comparison, the welfare cost of inflation obtained from the semi-log non-linear long-run money-demand function, derived using the PPR method, for 0–10 % of inflation ranges between 0.4930 and 1.9468 % of GDP. However, the standard errors associated with the welfare cost estimates obtained from PPR relative to the linear models tend to indicate that the nonlinear money demand provides more precise estimates of the welfare costs primarily for higher rates of inflation.

Appendices

Appendix

See Table 8

Table 8 Johansen cointegration test results

The basics of projection pursuit method

PPR is an extension of additive models, and was developed by Friedman and Werner (1981). Like additive models, this model is helpful in reducing the curse of dimensionality in that it uses univariate regression functions, and not multivariate regression functions. Thus the estimation becomes much faster in terms of computation. But this model is different from additive models in that it first projects the data matrix of explanatory variables (usually denoted by $X)$ in the optimal direction before applying smoothing functions to these explanatory variables.

The model consists of linear combinations of non-linear transformations of linear combinations of explanatory variables. The basic model takes the form

$$\begin{aligned} Y=\beta _0 +\sum \limits _{j=1}^{m_1 } {\gamma _j f_j (\beta _j^{\prime } x)+\eta } \end{aligned}$$

(8)

where $x$ is a column vector containing a particular row of the design matrix $X$ which contains $p$ explanatory variables (columns) and $n$ observations (row). Here $Y$ is the a particular observation variable (identifying the row being considered) to be predicted, $\beta 0$ here is refers to the ridge coefficient, which is the usual intercept term in a usual regression function. $\gamma _{j}$ is the coefficient corresponding to $j$th projection directions, $\{ \beta j\}$ is a collection of $r$ vectors (each a unit vector of length $p$, since they are the projection vectors) which contain the unknown parameters. Finally $r$ is the number of modeled smoothed non-parametric functions to be used as constructed explanatory variables. The value of $m_{1}$ is found through cross-validation or a forward stage-wise strategy which stops when the model fit cannot be significantly improved. For large values of $m_{1}$ and an appropriate set of functions fj, the PPR model is considered a universal estimator as it can estimate any continuous function in R $p$. Thus this model takes the form of the basic additive model but with the additional $\beta j$ component; making it fit $\beta j^{\prime }x$ rather than the actual inputs $x$. The vector $\beta j^{\prime }x$ is the projection of $X$ onto the unit vector $\beta j$, where the directions $\beta j$ are chosen to optimize model fit. The functions fj are unspecified by the model and estimated using some flexible smoothing method; preferably one with well defined second derivatives to simplify computation. This allows the PPR to be very general as it fits non-linear functions $f_{j}$ of any class of linear combinations in $X$.

For a given set of data $(y_{i},x_{i})$, the goal is to minimize the squared error

$$\begin{aligned} S=\sum \limits _{i=1}^n {\left[ y_i -\sum \limits _{j=1}^{m_1 } {f_j (\beta _j^{\prime } x_i )} \right] ^{2}} \end{aligned}$$

(9)

over the functions $f_{j}$ and vectors $\beta _{j}$. After estimating the smoothing functions $f_{j}$, one generally uses the Gauss-Newton iterated convergence technique to solve for $\beta _{j}$; provided that the functions $f_{j}$ are twice differentiable. It has been shown that the convergence rate, the bias and the variance are affected by the estimation of $\beta _{j}$ and $f_{j}$. It has also been shown that $\beta _{j}$ converges at an order of $n^{1/2}$, while $f_{j}$ converges at a slightly worse order.