# Trends and Cycles in Economic Aggregates

Author: Andrew McCulloch

Economists tend to see the extent of economic growth measured over periods longer than 10 years as being due to factors such as skills, technology and innovation while short-term fluctuations, such as recessions, are due to economic shocks of various kinds. The paths followed by economic aggregates, such as the level of economic output or GDP, are often seen therefore as being composed of separate trend and cyclical components. The trend in an economic aggregate such as GDP identifies the long-term component of growth while the cycle measures short-term fluctuations. In this article I will look at one method for estimating trend and cycle components using quarterly UK GDP from 1955 Q1 to 2012 Q1 as an example. The importance of understanding the cyclical position of the economy will also be discussed.

In order to estimate trend and cycle components it is necessary to solve the following equation:

where Yt is the log of GDP at time t. Clearly with a single piece of data on the left of the equation and two unknowns on the right of the equation, a unique solution does not exist.

In order to calculate a solution we make two assumptions. Firstly, we assume that the cyclical component consists of temporary fluctuations with properties similar to those of the random error in a statistical model. The second assumption concerns the shape or functional form of the trend component. The most obvious assumption about the trend is to assume that it is a straight line. This line can be fitted using a technique known as least squares which minimises the sum of the squared residuals or the difference between the observed and predicted values of Y at time t. The figure below shows such a straight line fit to the UK GDP data (ONS series YBHA) and the figure on the right shows the residuals, or the value of Yt minus the fitted of Yt, at each time point.

In this approach, the residuals chart the temporary fluctuations in the state of the economy. The figure suggests therefore that the economic cycle has a period of somewhere around 15 years. A period of below average economic performance from 1960 to 1975 was followed by a period of above average performance from 1975 to 2000. The view among economists is, however, that a recession is expected on average around once every eight years. It therefore appears that we need a more flexible trend that tracks the data more closely than a straight line. We don't want a trend that fits the data too closely, however, because such a trend would tend not to fit future data very well.

One method that we can used to fit an appropriate trend is to add a penalty term to the least squares criterion which penalizes the overall wiggliness of the fitted line. That is we want a closer fit than a straight line but we also don’t want to overfit the data and end up with a fitted line that is too close to the original data. The wiggliness of a line can be measured using the overall curvature or the rate of change of the gradient of a line. The fitting criterion for this model therefore includes the additional term:

where the parameter phi controls how wiggly is considered too wiggly. In statistics, this function is termed a smoothing spline but in economics it is known as the Hodrick-Precott filter. Most studies based on quarterly data use a value of 1600 for phi and we will follow this practice.

The figure below plots the smoothed fit from the Hodrick-Prescott filter for the period from 1970 to 1990 and on the right the corresponding residuals. It is clear that the fitted trend is much closer to the actual path of GDP in comparison to the previous straight line fit, while the plot of the residuals suggests that the economy experienced relatively short recessions in the early 1960s, mid 1970s, 1979-1980 and in 2008. There was also a recession in the early 1990s which we don’t seem to have identified and we have an ‘extra’ period of below trend growth in the late 1980s, but otherwise our estimate of the cycle seems to accord fairly well to the accepted path of the economy over this period.

Why does this matter? Since around 1990s the Bank of England has followed an inflation target, adjusting interest rates to try to keep inflation somewhere around 2 percent. Understanding where the economy is along the economic cycle is important within this framework because it has implications for the prospects for inflation. If the path of GDP is below trend there is little risk of economic expansion giving rise to rapidly rising wages and prices because we can simply expand production. In contrast, if the economy is at full capacity an increase in demand results in more people chasing the same bundle of goods so there is a tendency for prices and then wages to rise sharply. The estimates of trend and cycle at the end of the sample are unfortunately unreliable because they are unduly influenced by the latest data points.

At present, however, UK GDP is approximately at the same level it was in 2008 and it would seem likely that there is spare capacity in the economy but a shortage of demand. Government attempts to stimulate growth through programmes such as quantitative easing seem unlikely to cause major inflation.

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