Marcus Buckmann, Galina Potjagailo and Philip Schnattinger
Understanding the origins of currently high inflation is a challenge, since the effects from a range of large shocks are layered on top of each other. The rise of UK service price inflation to up to 6.9% in April might potentially reflect external shocks propagating to a wider range of prices and into domestic price pressures. In this blog post we disentangle what might have contributed to the rise in service inflation in the UK using a neural network enhanced with some economic intuition. Our analysis suggests that much of the increase stems from spillovers from goods prices and input costs, a build-up of service inflation inertia and wage effects, and a pick-up in inflation expectations.
Linear models can face limitations in explaining large, unprecedented fluctuations in inflation. At the same time, machine learning tools have become increasingly popular in forecasting and monitoring inflation and GDP growth. Such models can accommodate complex non-linearities and a larger number of variables, which makes them particularly appealing during periods of swift turning points and large shocks, and when exploring granular data. We employ a neural network Phillips curve model proposed in a recent working paper by Philippe Goulet Coulombe. The method allows us to extract signals for inflation from many variables, fed into the model according to the familiar building blocks of the Phillips Curve. Whereas the working paper also proposes a strategy to separate out a Phillips curve slope from an output gap estimate, our focus lies in the contributions from different Phillips curve components to the forecast and the signals that the model exploits over time rather than a structural identification.
An economically intuitive neural network
We use a neural network to approximate the Phillips Curve. Neural networks learn to recognise patterns in large data sets and make predictions. They pass data through several layers of interconnected nodes, where each connection between nodes is a parameter that is learned from data. For example, when predicting inflation, the parameters are calibrated to minimise the difference between the model’s final prediction of inflation and observed inflation. Most of the recent advances in artificial intelligence, such as ChatGPT, are based on huge neural networks with billions of parameters.
The model we use is of relatively small size, but still exploits a large set of variables and is able to learn a rich array of non-linearities. Rather than letting the model be purely driven by data, we impose economically motivated reduced-form assumptions by feeding the network with data split into sub-categories. Each sub-network learns to distil the information contained in the input variables and their non-linear interlinkages into a final neurons, or ‘latent’ components relevant for forecasting service inflation. The components represent the building blocks of a Phillips curve (Chart 1), parcelling up the drivers of inflation into past inflation dynamics, inflation expectations, a domestic output gap, and international prices to account for global price pressures affecting inflation in an open economy. For past domestic inflation, we further distinguish between a) domestic input costs and goods price inflation to reflect indirect effects from goods prices into services via for instance input-output linkages, and b) nominal inertia that captures past service price inflation, output costs and pay growth. Each component is derived from a set of aggregate (eg: unemployment rate, total service inflation) and disaggregate series (eg: two-digit industry output), also including lags and moving average transformations of each series. For example, rather than using one particular output gap measure, we use a range of real activity indicators that are likely to contain information about the output gap. The sum of the latent components forms the model’s overall inflation forecast.
Chart 1: A neural network with Phillips Curve structure
We estimate the model on a quarterly basis over the period 1988 until the first quarter of 2023, and after training over the first 12 years of the sample using the out-of-bag approach, we then run predictions in an out-of-sample exercise. We evaluate the model out-of-sample and re-calibrate the model every quarter to incorporate incoming data.
Service inflation forecast decomposition into Phillips curve contributions
The model forecasts service inflation relatively closely, with a good out-of-sample performance. Chart 2 decomposes the forecast into contributions from Phillips curve components. The forecast (black line) reproduces the main fluctuations in service inflation (grey dashed line in the left panel) over the sample period such as the surge during the late 1980s, the decline during the Covid-19 pandemic and the recent rise. According to the model, the surge of inflation during the end-1980s was associated with elevated inflation expectations (light and dark purple bars) and domestically generated inflation (light and dark green bars). After the subsequent policy interventions and establishment of inflation targeting, all contributions come down and the role of inflation expectations is diminished for the rest of the sample period.
Chart 2: Decomposition of service inflation via the neural Phillips Curve
Notes: 1-q ahead out-of-sample forecasts from 2000, out-of-bag cross validation up to 1999. Quarterly growth rates are annualised, with quarterly fluctuations smoothed out. Contributions to forecast (solid line) from PC components, relative to the mean of service inflation of 3.3% during 1997–2019. Dashed grey lines shows service inflation outturn. Left: zoomed in decomposition since 2020 Q1. 2023 Q2: current forecast period.
Throughout, we observe spill-overs from tradable goods into service inflation via input-output linkages and cost-push effects (light green bars), in particular during the 2000s and in 2014–15, yet these have typically not translated to inertia in service prices and pay growth (dark green bars). The effects from good prices and input costs were weak during the pandemic, reflecting that service inflation was falling but goods price inflation was being pushed up by supply shortages. International prices (blue bars) contribute only little overall, which is in accordance with service inflation being a measure of domestic price pressures not directly affected by terms of trade shocks (we find stronger effects for headline inflation).
The intrinsic dynamics of service prices and pay growth, or nominal inertia, have shifted infrequently in the past. During the decade following the financial crisis, nominal inertia (dark green bars) dragged on service inflation, likely due to timid wage growth and weak dynamics of service inflation during that period. But this has now reversed with the most recent rise in services prices. Since the beginning of 2022 the model detects jointly increased contributions from spillovers from input cost and from nominal inertia. These two contributions have been the largest at play. Since mid-2022, the inflation expectations contributions (light and dark purple bars) have also picked up, but the size of the effect remains relatively small compared to the early episode in the early 1990s, suggesting that expectations remain anchored. The output gap effect has contributed slightly positively, but the size of this effect has not grown since the initial post-pandemic recovery.
Recently strong synchronised signals from input costs and wage growth
We further dissect the signals that the model exploits from each input variable over time via Shapley value heat maps, derived from the out-of-sample exercise since 2000 (Chart 3). Apart from the year 2008, signals from variables related to goods prices and input costs have rarely been both as strong and synchronised as in the recent period (Panel a). And during earlier periods, positive signals from input and energy costs did not translate into rises of nominal inertia (Panel b). Apart from some variation in output and accommodation prices, all signals in the nominal inertia component were jointly pointing to below-mean service inflation, with the strongest negative signals stemming from average weekly earnings and lagged service dynamics. Since early 2021 this started to shift, initially via positive signals from earnings, followed by accommodation and catering, two industries where prices recovered post-pandemic. Since the second half of 2022, all nominal inertia signals have been synchronised and clearly positive.
Chart 3: Signals to the service inflation forecasts over time
Notes: Signals are derived as Shapley values for each quarter and variable. Darker red indicates stronger positive signal, darker blue indicates stronger negative signal. Indicators with stronger average signals are ordered on top, although relevance can shift over time. Panel c: Shapley values from additional 11 sectoral output series not shown for readability.
According to the model, the inflation-relevant output gap has contributed only slightly to inflation across most of the sample period (Panel c). During the global financial crisis in 2009, the model reads mostly positive signals and misses the fall in service inflation, albeit the Bank of England’s output gap series itself provided a negative persistent signal. Challenges in empirically detecting a strong Phillips curve relationship are not new and might be related to identification issues around the role of monetary policy. During the pandemic, our model detects a strengthened role of the output gap, with drag from the output gap explaining much of the fall of service inflation. Signals also became temporarily more dispersed, in line with the heterogeneous nature of the Covid-19 shock. Recently, signals have been mostly positive, but not very pronounced.
Summary and implications
The neural network Phillips curve model suggests that the recent rise in UK service inflation has been associated with a rise in nominal inertia related to lagged service inflation dynamics and pay growth. This component has fluctuated little and represented a drag on service inflation in the past. Its rise could reflect second-round effects affecting domestic inflation beyond the direct effects from external shocks, although the degree to which we can distinguish between the two in a non-structural model is limited. As in any empirical model, the number of observations following high inflation is limited, and so uncertainty around model predictions for these periods is higher.
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