Lucas’s Two Illustrations of the Quantity Theory and Whiteman’s Critique

Lucas’s Two Illustrations of the Quantity Theory and Whiteman’s Critique#

Note

This section describes Robert E. Lucas, Jr., “Two Illustrations of the Quantity Theory of Money,” American Economic Review 70(5), 1005–1014 (1980); Charles H. Whiteman, “Lucas on the Quantity Theory: Hypothesis Testing without Theory,” American Economic Review 74(4), 742–749 (1984); and Thomas J. Sargent and Paolo Surico, “Two Illustrations of the Quantity Theory of Money: Breakdowns and Revivals,” American Economic Review 101(1), 109–128 (2011). We follow their arguments and omit the details of the Bayesian estimation. A computational treatment, with Python code for the filters, the scatter plots, the VAR, and the estimation of the DSGE model, is the QuantEcon lecture Two Illustrations of the Quantity Theory of Money.

This is the third application in this book of Sims’s approximation error formula (385), after Seasonality and Approximation Errors and Money Demand in Hyperinflations: A Misspecified Regression and Sims’s Approximation Formula.

Lucas’s method#

Lucas (1980) set out to illustrate two central quantity-theoretic propositions: a change in the growth rate of money induces (i) an equal change in the rate of inflation, and (ii) an equal change in the nominal interest rate. Rather than estimate a structural model — a course he judged to require “nesting the two hypotheses in question within a complex maintained hypothesis, which must be accepted as valid in order to carry out the test” — he applied a deliberately atheoretical two-step procedure.

Step 1: filter. Replace each series by a two-sided, exponentially weighted moving average. For money growth \(\mu_t\),

(503)#\[\bar\mu_t(\beta) \;=\; \frac{1-\beta}{1+\beta}\sum_{k=-\infty}^{\infty}\beta^{|k|}\mu_{t-k}, \qquad 0<\beta<1,\]

the weights being normalized to sum to one. Step 2: scatter. Plot the filtered inflation–money pairs \((\bar\mu_t(\beta),\bar\pi_t(\beta))\) and the filtered interest–money pairs \((\bar\mu_t(\beta),\bar\rho_t(\beta))\) for a range of \(\beta\), and look at the slope. For US data over 1955–1975 and \(\beta\) near \(1\), both scatters hugged the \(45^\circ\) line: Lucas read \(a_1\approx a_2\approx 1\) in

(504)#\[\bar\pi_t(\beta) = a_1\,\bar\mu_t(\beta)+\varepsilon_{1t}, \qquad \bar\rho_t(\beta) = a_2\,\bar\mu_t(\beta)+\varepsilon_{2t},\]

as evidence for the two quantity-theory propositions, and hence against the Mundell–Tobin effect, by which higher inflation lowers the return on money, shifts portfolios toward real capital, and makes the nominal interest rate rise less than one-for-one with money growth.

Using the filter kit of this chapter, the transfer function of (503) is

(505)#\[B(e^{-i\omega}) \;=\; \frac{(1-\beta)^2}{1+\beta^2-2\beta\cos\omega}.\]

Note that \(B(1)=1\) for every \(\beta\) — the filter has unit gain at frequency zero — while for any fixed \(\omega\ne0\), \(B(e^{-i\omega})\to0\) as \(\beta\to1\). So (503) is a low-pass filter that, as \(\beta\uparrow1\), concentrates all of its power at \(\omega=0\). Filtered money growth has spectral density \(\lvert B(e^{-i\omega})\rvert^2 S_{\mu\mu}(e^{-i\omega})\), which for \(\beta\) near one is massed almost entirely at the origin.

Whiteman’s first point: what the method measures#

Whiteman’s (1984) first contribution was to say precisely what Lucas’s graphical procedure estimates. Consider the two-sided population projection of the nominal interest rate on the entire money-growth process,

(506)#\[\rho_t = \sum_{k=-\infty}^{\infty}\gamma_k\,\mu_{t-k} + \eta_t, \qquad E\mu_{t-k}\eta_t = 0 \ \ \forall k,\]

whose transfer function is the familiar ratio of the cross spectrum to the spectrum (The Cross Spectrum, The Effects of Filtering on One-Sided and Two-Sided Projections),

(507)#\[\gamma(e^{-i\omega}) = \frac{S_{\rho\mu}(e^{-i\omega})}{S_{\mu\mu}(e^{-i\omega})} .\]

Equation (506) cannot be estimated — it has infinitely many parameters — so any practitioner fits some restricted lag distribution \(\gamma'\). By Sims’s approximation-error formula (Exercise 33), least squares in population chooses \(\gamma'\) to minimize the spectral-density-weighted distance

(508)#\[\int_{-\pi}^{\pi}\big|\gamma(e^{-i\omega})-\gamma'(e^{-i\omega})\big|^{2}\, S_{\mu\mu}(e^{-i\omega})\,d\omega .\]

Now apply (508) to Lucas’s procedure. Regressing one filtered series on another with a single coefficient \(a_2\) is the extreme restriction \(\gamma'(e^{-i\omega})\equiv a_2\), a constant lag-distribution transfer function; and the relevant weighting density is that of filtered money growth. So Lucas’s method chooses \(a_2\) to minimize

(509)#\[\int_{-\pi}^{\pi}\big|\gamma(e^{-i\omega})-a_2\big|^{2}\, S_{\bar\mu(\beta)}(e^{-i\omega})\,d\omega .\]

Because the weighting density is massed at \(\omega=0\) when \(\beta\) is near one, essentially all of the weight falls on getting the approximation right at the origin. Hence

(510)#\[a_2 \;\longrightarrow\; \gamma(e^{-i\cdot 0}) \;=\; \gamma(1) \;=\; \sum_{k=-\infty}^{\infty}\gamma_k \;=\; \frac{S_{\rho\mu}(0)}{S_{\mu\mu}(0)} \qquad\text{as }\beta\to1 .\]

Lucas’s scatter-plot slope is an estimate of the sum of the coefficients in a two-sided distributed lag regression — equivalently, the ratio of the cross spectrum to the spectrum at frequency zero. Sargent and Surico write this object \(h(0)\); it is a population magnitude that any time series model delivers. Whiteman emphasized that this holds “regardless of the time-series properties” of the two series: the method is a robust, if extremely roundabout, estimator of a sum of lag coefficients.

Two features of the book’s machinery bear on this. Filtering both series with the same filter is legitimate because a common filter applied to \(y\) and \(x\) leaves the two-sided projection unaltered, as The Effects of Filtering on One-Sided and Two-Sided Projections establishes; a common filter disturbs the one-sided projection, except when Granger non-causality makes the two coincide. Lucas’s low-pass filter therefore reweights which frequencies the fit attends to without changing the \(\gamma_k\) being estimated. And the exercise is a benign cousin of the Slutsky–Kuznets effects: filtering can manufacture apparent regularities, so one must know what a filter does before reading economics off a smoothed plot.

Why not just add up estimated lag coefficients?

Whiteman’s footnote on this point is a small gem of projection theory, and it explains why Lucas’s roundabout method is actually the safer one.

Suppose that instead of Lucas’s procedure you fit a parsimonious rational lag distribution \(\gamma'(e^{-i\omega})\) by least squares and then report \(\gamma'(1)\). The trouble is that the sum of coefficients is not a continuous function of the least-squares criterion (508). The criterion is an integral, and the single frequency \(\omega=0\) is a set of measure zero: you can change \(\gamma'(1)\) arbitrarily — even send it to infinity — while barely moving the value of the integral, i.e. without materially changing the fit.

Whiteman illustrated this with Mills’s (1982) estimated transfer function. Perturbing one estimated coefficient from \(-0.56\) to \(-0.58\) — a change amounting to less than a tenth of one standard error, and invisible in the fit — drives the denominator of the rational lag distribution toward zero at \(z=1\), so that \(\gamma'(1)\) swings from unity to \(+\infty\). The reported standard error on the sum is therefore badly misleading.

Lucas’s method escapes this trap because it estimates the sum by fitting a constant, and a constant transfer function is continuous with respect to the least-squares metric. The price is that Lucas’s procedure deliberately sacrifices accuracy about individual lag coefficients in order to buy accuracy about their sum — and, as Whiteman notes, the same “controlled” approximation error that makes the sum well estimated destroys any hope of attaching a standard error to it.

Whiteman’s second point: what the measurement means#

Knowing that Lucas measured \(\gamma(1)\) still leaves the substantive question: does \(\gamma(1)\approx1\) constitute evidence against the Mundell–Tobin effect? To answer, Whiteman did what Lucas had declined to do — he adopted an explicit structural model and computed the population version of Lucas’s statistic inside it. Tellingly, he chose Lucas’s own (1975) equilibrium model of the business cycle, which is convenient because it isolates the Mundell–Tobin effect in a single parameter.

In that model, log output and the log real return to capital are \(y_t=\delta_0'+\delta_1'k_t\) and \(r_t=\delta_0-\delta_1k_t\); the demands for capital and for real balances are

(511)#\[\begin{split}\begin{aligned} k_{t+1} &= \alpha_0' + \alpha_1 E_t r_{t+1} + \alpha_2\,(E_t p_{t+1}-p_t) + \alpha_3 k_t,\\ m_t - p_t &= \beta_0' - \beta_1 E_t r_{t+1} - \beta_2\,(E_t p_{t+1}-p_t) + \beta_3 k_t, \end{aligned}\end{split}\]

with money supply \(m_t = A(L)e_t\) for a fundamental \(\{e_t\}\). The parameter \(\alpha_2\) is the Mundell–Tobin effect: when \(\alpha_2=0\) the demand for capital is independent of expected inflation, monetary disturbances leave the steady-state capital stock alone, and the second equation of (511) reduces to a version of Cagan’s portfolio-balance schedule — the very schedule studied in Money Demand in Hyperinflations: A Misspecified Regression and Sims’s Approximation Formula and Some Applications to Rational Expectations Models, whose forward solution \(p_t=\frac{1}{1+\beta_2}\sum_{k\ge0}\big(\frac{\beta_2}{1+\beta_2}\big)^k E_t m_{t+k}\) has lead weights summing to one. In that case the model does exhibit the two quantity-theory propositions.

Two pieces of the book’s apparatus organize the solution. Whiteman assumed that \((k_{t+1},p_t)\) fails to Granger-cause \(m_t\); by Sims’s theorem this makes the moving average representation triangular,

(512)#\[\begin{split}\begin{bmatrix} m_t \\ k_{t+1} \\ p_t\end{bmatrix} = \begin{bmatrix} A(L) & 0 & 0\\ B(L) & D(L) & G(L)\\ C(L) & F(L) & H(L)\end{bmatrix} \begin{bmatrix} e_t \\ w_t \\ v_t\end{bmatrix},\end{split}\]

exactly the device used in Money Demand in Hyperinflations: A Misspecified Regression and Sims’s Approximation Formula and A Difficulty in Interpreting Vector Autoregressions. Rational expectations then determine \(B(L)\) and \(C(L)\) uniquely from the money-supply rule \(A(L)\), by the familiar factorization into stable and unstable roots and the geometric-lead formulas of this chapter. The transfer function of the inflation-on-money-growth projection turns out to be simply \(\gamma^{\pi}(z)=C(z)/A(z)\).

Whiteman’s conclusions are these.

  • The first illustration is uninformative about \(\alpha_2\). When \(\alpha_2=0\), \(\gamma^\pi(1)=1-\frac{\beta_2}{1+\beta_2}A\big(\frac{\beta_2}{1+\beta_2}\big)A(1)^{-1}\), so \(\gamma^\pi(1)=1\) requires \(A(1)^{-1}=0\), i.e. a random walk component in the money supply — the condition Lucas had himself employed in his 1972 and 1975 papers. But when that condition holds, \(\gamma^\pi(1)=1\) whatever the value of \(\alpha_2\). Lucas’s first result is essentially independent of the magnitude of the Mundell–Tobin effect.

  • The second illustration points the other way. Conditioning on \(\gamma^\pi(1)=1\), the requirement \(\gamma^\rho(1)=1\) becomes a restriction involving all the model’s parameters together with those of the money-supply rule; it implies \(\alpha_2=0\) only if \(A^*(\lambda_2^{-1})/A^*(1)=0\), which needs either a random walk in money growth or the absence of an autoregressive representation. Neither is plausible. Moreover Lucas’s own figures argue against the first: were the money spectrum already infinite at \(\omega=0\), raising \(\beta\) (which enhances low-frequency power) would leave \(a_1\) and \(a_2\) unchanged — yet Whiteman’s replication found \(a_1\) moving from \(0.02\) to \(0.08\) to \(0.87\) to \(0.99\) as \(\beta\) ran through \(0,\,0.5,\,0.9,\,0.95\). The most plausible reading of the evidence taken as a whole is that the restriction holds with \(\alpha_2\ne0\) — that is, Lucas’s numbers can be read as evidence in favor of the very Mundell–Tobin effect he took them to refute.

Lucas’s critique returns here against Lucas’s own paper. Reduced-form low-frequency correlations are complicated functions of the structure of the economy and of the laws of motion of the processes agents care about; they are unlikely to reveal much about the true model “unless one already has it very much in mind.” This is the same non-invariance that drives Some Applications to Rational Expectations Models, A Difficulty in Interpreting Vector Autoregressions, Exact Linear Rational Expectations Models, Chapter XIV and — in the guise of a misspecified regression whose weighting spectrum is generated by the policy rule — Money Demand in Hyperinflations: A Misspecified Regression and Sims’s Approximation Formula. Whiteman’s title says it: beware of hypothesis testing without theory.

Sargent and Surico: breakdowns and revivals#

Sargent and Surico (2011) take up Whiteman’s reformulation and push it in a quantitative direction, asking when Lucas’s unit slopes “obtain in reality” and when they “break down” — a caveat Lucas himself had attached to his propositions. Their organizing object is exactly Whiteman’s,

(513)#\[h(0) \;=\; \frac{S_{yz}(0)}{S_{z}(0)},\]

which any state-space model delivers. If \(X_{t+1}=AX_t+BW_{t+1}\), \(Y_t=CX_t\) with \(A\) stable, then writing the long-run multiplier \(G=C(I-A)^{-1}B\) gives \(2\pi S_Y(0)=GG'\) and hence \(h_{y,z}(0)=[GG']_{yz}/[GG']_{zz}\) — a one-line computation from the compact state-space notation of this chapter.

Their findings:

  • The slopes are unstable. Extending the data to 1900–2005 and computing Lucas’s regressions on filtered data, the inflation-on-money-growth slope is \(1.13\) for 1900–1928, \(0.39\) for 1929–1954, \(0.86\) for Lucas’s own 1955–1975, and \(-0.03\) for 1984–2005; the interest-rate slope is \(0.62\) for 1955–1975 but \(0.06\) or negative in several other subperiods. Lucas’s sample turns out to be atypical. A VAR with time-varying coefficients and stochastic volatility tells the same story: the posterior median of \(h(0)\) drifts substantially, peaking in the 1970s and reaching its lowest values in the 1940s and in the most recent twenty years.

  • Monetary policy accounts for the instability. Estimating a New Keynesian DSGE model over 1960–1983 reproduces Lucas’s unit slopes (implied \(h_{\pi,\Delta m}(0)\approx1.01\), \(h_{R,\Delta m}(0)\approx0.82\)). Then, freezing every nonpolicy parameter and varying only the policy-rule coefficients, the implied \(h(0)\)’s sweep across essentially the whole range from \(0\) to \(1\) — covering the entire span of the empirical estimates. The low-frequency slopes are not policy invariant.

  • The economics of the breakdown. A policy rule that responds aggressively to incipient inflation prevents persistent movements in money growth from emerging and thereby eradicates the unit slopes; a rule that responds too weakly, acceding to persistent money growth, revives them. As the authors put it, Lucas’s illustrations “come back” precisely when a monetary authority allows persistent movements in money growth — so, as citizens, they prefer the times when the propositions break down.

This is Whiteman’s argument made operational and quantitative: the same statistic, computed inside a fully articulated model, moves with the money-supply rule. The frequency-zero slope measures a genuine feature of the data, but what it means depends on the policy regime that generated the money process — which is, once more, the weighting-spectrum lesson of Sims’s approximation formula and the substantive lesson of the Lucas critique.

Exercise 32

Let \(\mu_t\) have spectral density \(S_{\mu\mu}\), and let \(\bar\mu_t(\beta)\) be the Lucas filter (503) with transfer function (505).

A. Verify that \(B(1)=1\) for all \(\beta\in(0,1)\), and that \(B(e^{-i\omega})\to0\) as \(\beta\to1\) for each fixed \(\omega\ne0\).

B. Suppose \(y_t=\gamma(L)\mu_t+\eta_t\) as in (506). Show that the population regression coefficient of \(\bar y_t(\beta)\) on \(\bar\mu_t(\beta)\) is a weighted average of \(\gamma(e^{-i\omega})\), and identify the weights. Conclude that the coefficient \(\to\gamma(1)\) as \(\beta\to1\).

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