What a Puzzle! Unravelling why UK Phillips Curves were Unstable

Between 1860 and 2021, UK Phillips curves linking wage inflation ( ∆ w ) and unemployment ( U r ) exhibit every slope in sub-period regressions from strongly negative, slightly negative, flat, slightly positive and strongly positive. These sub-period outcomes are predicted by an econometric model of real-wage growth expressed in terms of ∆ w . Correcting ∆ w for its regressors other than U r , its sub-period regressions on U r all have the same negative slope. However, ‘shifts’ in the real-wage model’s variables do not explain the instabilities: surprisingly, the Phillips curves shift when some of the real-wage model’s sub-period regressors are insignificant.


Introduction
The instability over time in Phillips curves is well known and well documented by both academics and policy-makers; see Del Negro et al. (2020) and Haldane and Quah (1999) for the former and Powell (2019) and Cunliffe (2017) for the latter, both from a US and UK perspective.This instability is demonstrated for the UK in Figure 2 which records five subsample estimates of the Phillips curve using annual data from 1860-2021. 1 Every slope in sub-period relationships between wage inflation (∆w, where lower case letters denote logs) and the unemployment rate (U r ) is observed, from strongly negative, slightly negative, flat, slightly positive and strongly positive.These outcomes are compared with those over the same sub-samples derived from the econometric model of real-wage growth in Castle and Hendry (2014) updated to 2021 in Castle et al. (2023), expressed in terms of ∆w, revealing a close match in every sub-period.Thus, the constant real-wage growth model can account for the instabilities in the simple Phillips' curve, a successful mis-specification encompassing result (see Hendry, 1995, Ch. 14).
We first compare Phillips Curves in price and wage inflation over the original sample in Phillips (1958), then extend the estimates to 2021 to establish its well-known instability.Next, we derive the nominal-wage inflation-unemployment relation from the real-wage model in Castle et al. (2023).We use that transform to demonstrate that its fitted values ∆w t closely replicate the shifts for every sub-period.Thus the additional regressors must explain the shifts. 2Confirmation that the underlying relationship between ∆w t and U r,t is constant conditional on the other regressors is shown by the relation between ∆w t and U r,t derived from the transformed real wage equation having essentially the same slope in every sub-period.However, it transpires that it is not 'shifts' in the real-wage model's regressors that explain the instabilities, which is a puzzle.
The structure of this paper is as follows.Section 2 compares Phillips Curves in price and wage inflation and replicates the original Phillips (1958) Phillips (1958) related changes in nominal wages to the unemployment rate as both are labour market variables, but many recent variants use price inflation (see Forder, 2014 andHoover, 2015, for historical perspectives).Consequently, Figure 1 compares nominal wage and price inflation using annual observations on his data period 1860-1913.Phillips defined wage inflation, Dw, as Dw = 0.5(W t+1 − W t−1 )/W t , whereas we use the standard definition of the change in the log (there is a difference in timing of the peaks and troughs, but the two series are highly correlated: see Hendry, 2001).Phillips was aware that discrete approximations created moving-average errors (see Phillips, 2000), but in 1958 these were nearly impossible to estimate.He also knew that the 'loops' around his long-run relation represented dynamic adjustments, so calculated his equation from subsets of unemployment levels within which the average over a business cycle should be close to zero (see Desai, 1975).
∆p t is price inflation measured by the GDP deflator.As Figure 1 shows, a cubic spline fitted to ∆w matches Phillips' non-linear form, whereas price inflation results in a nearly straight line.Thus, all our results relate to wage inflation.Note that since Phillips conducted his study, pre-World War I (WWI) data on unemployment have been substantially revised by Boyer and Hatton (2002), but our pre-WWI results are close to those Phillips reported.
However, the simple bivariate relation between ∆w and U r was not to last as shown in Figure 2. The five sub-periods plots of ∆w against U r are chosen as the original Phillips' period 1860Phillips' period -1913 (before lags) (before lags); WWI to the end of WWII;1946-1980, namely   That the Phillips Curve did not fail during the sub-sample which included WWI, the inter-war period with the post-war collapse and then the 'Great Depression', plus World War II is surprising (although the downward slope estimate doubled), but sloping up during the two less turbulent post-WWII periods, and flat since 2012 while U r varied over 4%-8%, seem less excusable and confirms the unstable relation of ∆w to U r .The 'outliers' from wars, oil crises, price controls, indexation and the 'Great Depression' are highlighted in 'boxes' as they derive from different extraneous causes at different times.The resulting sub-sample coefficient estimates { b} in the regression ∆w t = a + bU r,t , with their heteroscedasticity and autocorrelation consistent (HAC) standard errors (see Andrews, 1991)  While Figure 1 shows cubic splines graphically fitted to the data given the non-linear relation in Phillips (1958), these are close to the linear sub-sample estimated regressions in Figure 2, confirming coefficients of U r change substantially over time.

Deriving a wage inflation-unemployment relation from a real-wage model
The natural explanation of such unstable estimates is that relevant variables not included in the simple model experience shifts.If all excluded variables were stationary and maintained a constant correlation with U r , its coefficient would be constant despite the omissions.Conversely, if the additional regressors included in the econometric model of real-wage growth in Castle et al. (2023) (reported for 1862-2021 in ( 1)) explained the shifts, its outcomes predicted for ∆w t in every subperiod should provide a close match.Figure 3 adds to Figure 2 the resulting sub-period plots of the full-sample { ∆w t } against U r,t and the close match confirms that the regressors in (1) model account for the instabilities in the simple Phillips' curve.Since ( 1) is constant over the whole period T= 1862-2021 it successfully mis-specification encompasses the shifting Phillips curves.
34 F nl (24, 121) = 1.09Coefficient standard errors are in parentheses (HAC in brackets), σ is the residual standard deviation, F ar tests residual autocorrelation (see Godfrey, 1978), F arch tests autoregressive conditional heteroscedasticity (see Engle, 1982), F het tests residual heteroskedasticity (see White, 1980), χ 2 nd (2) tests non-Normality (see Doornik and Hansen, 2008), F reset tests non-linearity (see Ramsey, 1969), F nl also tests non-linearity (see Castle and Hendry, 2010), and F chow tests parameter constancy (see Chow, 1960).One star indicates test significance at 5%, two at 1%.In (1), ∆ (y − l) t measures labour productivity and the labour share of income is given by (w − p − y + l) t where µ is its sample mean.S xxxx is a step indicator taking the value 1 till the date xxxx and 0 after, and I xxxx is an indicator variable taking the value 1 for that observation only.
is a logistic smooth transition function (see Luukkonen et al., 1988) where the scaling bounds the function between [−1, 0]). 3ost recently, ( 1) is constant over Brexit, the pandemic and the UK government's furlough scheme during lockdowns, and also passes a test for super-exogeneity of its contemporaneous regressors (see Engle et al., 1983 andEngle andHendry, 1993).Expressing (1) in terms of ∆w t in relation to U r,t plus other drivers shown as [•] yields: The coefficient of ∆p t is carried over at unity from undoing ∆(w − p) t , so 0.86 = 1 − 0.14 from 1), but is 0.96(0.08)if estimated unrestrictedly on the right-hand side of (1).

Sub-period implications
To evaluate if correcting for the additional drivers produced stable subsample estimates, we calculated x t as the sum of all the influences on ∆w t in [•] other than U r,t in (3) to derive (∆w t | x t ) as the residuals from the full-sample regression of ∆w t on x t , shown in ( 4) with HAC standard errors.
x t (4) Thus, wage inflation is only corrected by a scalar which uses the same coefficients in all subperiods, leading to the full-sample regression recorded in (5).The resulting sub-period regressions are shown in Figure 4 and are nearly identical across all sub-periods, with the full-sample regression in (5).Although the Frisch and Waugh (1933) theorem suggests that U r,t should also be corrected for x t , it was essentially orthogonal to x t .This was a further surprise that the composite variable that explained most of the variance of real wages was not related to U r,t .However, as shown in Table 3, while U r,t was uncorrelated with x t on the full sample, U r,t was significantly correlated with x t in those sub-samples where the Phillips curve shifted.The plot thickens...In fact, the unconditional regression of ∆w t on U r,t delivers a similar coefficient of −0.72 to ( 5 Without accounting for price inflation, labour productivity, the wage share, non-linear wageprice spiral effects and exogenous shocks such as wars and oil crises, the Phillips curve is nonconstant.These additional drivers obscure a constant nominal wage-unemployment rate relation over the last 160 years shown in Figure 5(b).

Testing the validity of conditioning on U r,t
Valid conditioning in non-constant processes requires super exogeneity, see Engle et al. (1983) and Engle and Hendry (1993).Extending the automated test in Hendry and Santos (2010), we first apply impulse indicator saturation (IIS) and step indicator saturation (SIS) jointly at a significance level of 0.1% to the unconditional U r,t regression on a constant, shown in (7) to select step shifts over 1862-2021 to match the estimation samples above which were shorter from lagged variables.
In both ( 7) and ( 8), the standard errors reported are HAC, but the conventional standard errors also confirm significance as do the F exclude tests of excluding the indicators.Importantly, none of the step indicators in ( 7) also enter (3).
The five selected indicators are highly significant when added to (4) as reported in (8), which is expected as they are a 'big effects proxy' for the missing role of U r,t .
x t − 0.04 However, all the step indicators become insignificant when added to (5) as shown in (9) and no diagnostic tests reject.Thus the major shifts in U r,t do not enter the regression of (∆w t | x t ) on U r,t confirming it is super exogenous in that model.
These results are consistent with the evidence in Castle and Hendry (2014) that most UK unemployment has been involuntary.Nevertheless, ( 5) and ( 9) are projections from a multivariate relation determining real wages, where the nominal level is determined by the price equation as in Hendry (2001).
6 Subsample equations: what caused the shifts?
Figure 6 plots the time series of ∆w t and U r,t with the sub-periods shown, where the bars mark the two world wars.Their patterns within each sub-period are very different, so it is unsurprising that the original Phillips curve would not be constant across the five subsamples, as shown in Figure 2.
Table 4 records the regression coefficient values with |t| ≥ 2 from fitting the general model to subsamples to examine what changes were due to which variables (e t−2 = (w −p−y −l − µ) t−2 ).The first two sub-periods are similar to the full sample, but the next two differ in many respects (the final sample is too short to be reliable).In particular, the impacts of (U r,t − 0.05) = Ūr,t and ( Ūr,t ) 2 are then insignificant, as are the non-linear inflation reactions.These absent impacts on wage inflation explain the upward slopes.Imposing the whole sample coefficient estimates for the zero values over 1946-1980 in Table 4 yields an equation standard error of σ = 1.17% as against the unrestricted fit of σ = 1.11%.Similarly, for 1981-2011, σ = 0.91% versus σ = 0.98% (the lower constrained value is an artefact of not counting restricted coefficients in the degrees of freedom).In neither case were any mis-specification tests significant for the constrained models, consistent with the overall constancy of the general model despite subsample estimate variations.Thus, the slope changes in the original Phillips curve are due to the lack of variability of the real-wage model regressors in the sub-samples: its stability needed hidden co-breaking (Hendry and Massmann, 2007) between the variables in the real-wage model, and the absence of some impacts stopped that occurring.

Conclusion
The UK Phillips' curve relating changes in the log of nominal wages to unemployment is unstable.Sub-period relationships between wage inflation (∆w) and unemployment (U r ) can be strongly negative, slightly negative, flat, slightly positive and strongly positive in a time series from 1860 to 2021.In Section 3, mis-specification encompassing (see Hendry and Nielsen, 2007, Ch. 13) revealed that the shifts in the subsample Phillips curves could be accounted for by a constant parameter real-wage equation; In Section 4, partialling out from nominal wages the full-sample estimated coefficient linear combination of the regressors x t , other than unemployment, showed that the resulting subsample equa-tions had essentially the same downward slopes of between −0.67 to −0.85;In Section 5 the validity of conditioning (∆w t | x t ) on U r,t was confirmed; In Section 6, the insignificance of estimated coefficients in subsample real-wage models in Table 4, matched when the Phillips curve shifted, as did the significance of the correlation of U r,t with x t in subsamples as in Table 3. Imposing the full-sample estimated values on insignificant subsample coefficients produced constant equations with no deterioration in fit, identifying the culprits behind the instability by their absence.Quite a surprise that the constancy of the nominal wage change-unemployment rate relationship depended on co-breaking of all the variables that it omitted from the constant real-wage model, then failed only when those lacked significance.
Although the whole sample regression of ∆w t on U r,t delivers the same coefficient as in a much more general constant parameter equation, it is not a useful way to model the inflationunemployment relation important to economic policy.Instead, useful policy implications require taking account of the constant parameter, multivariate, non-linear, dynamic relationship for real wages that encompasses the original Phillips curve, interacted with a price inflation model to determine the overall level and persistence of inflation, as in Castle et al. (2023).
non-linear relation of wage inflation to the unemployment rate.Section 3 derives the nominal-wage inflation-unemployment relation from the real-wage model in CHM and Section 4 analyzes its sub-period implications.Section 5 tests the validity of conditioning on U r,t .Section 6 examines subsample equations to ascertain what caused the shifts.Section 7 concludes and the Appendix 8 provides definitions and sources of the data series used. 2 Comparing Phillips Curves in price and wage inflation ∆w spline fit to U r showing dates ∆p spline fit to U r shown by circles 0to U r showing dates ∆p spline fit to U r shown by circles

Figure 1 :
Figure 1: Comparing Phillips Curve in ∆p with ∆w.
Four selected consecutive impulse indicators during WWII are combined as their coefficients were equal and opposite signed (I W W II = I 1942 + I 1943 − I 1944 − I 1945 ) and:

Figure 3 :
Figure 3: Comparing direct and derived Phillips Curves.

Figure 6 :
Figure 6: Time series of ∆w t and U r,t with the two World Wars shaded and the other subsamples shown by vertical lines.
post-war recovery till the end of the oil crisis; 1981-2011 which was the sample end inCastle and Hendry, 2014; and 2011-2021which includes Brexit, the Covid-19 pandemic lockdowns and the UK government's furlough scheme to prevent excessive unemployment.

Table 1 :
are shown in Table1.Estimates and HAC standard errors of b in the regression ∆w t = a + bU r,t

Table 2
reports the estimated coefficients of unemployment and their HAC standard errors, which stand in sharp contrast to the estimates in Table1.

Table 2 :
Coefficients of U r,t and HAC standard errors in the subsample (∆w t | x t ) regressions

Table 4 :
Coefficients with |t| ≥ 2 in the subsample regressions for the general model of ∆(w − p) t .
Such behavior prompted five puzzles: Puzzle 1: what caused the Phillips curve slopes to change so much?Puzzle 2: why is the Phillips curve over the very turbulent period 1914-1945 similar to the original over the relatively stable 1860-1913?Puzzle 3: can a constant-parameter real-wage model successfully encompass a shifting nominal wage equation?Puzzle 4: does the lack of correlation of U r with the full-sample estimated combination of the variables x t that explains changes in nominal wages, hold in subsamples?Puzzle 5: why does correcting ∆w t by x t produce a near constant subsample set of equations for ∆w t | x t on U r,t ?These puzzles concerning aspects of what caused the shifts in subsample Phillips curves can all be resolved as follows.