Which Indicators Matter? Analyzing the Swiss Business Cycle Using a Large-Scale Mixed-Frequency Dynamic Factor Model

Abstract

For policy institutions such as central banks, it is important to have a timely and accurate measure of past and current economic activity and the business cycle situation. The most prominent example for such a measure is gross domestic product (GDP). However, GDP is only released at a quarterly frequency and with a substantial delay. Furthermore, it captures elements that are not directly linked to the business cycle and the underlying momentum of the economy. In this paper, I construct a new business cycle index for the Swiss economy, which uses state-of-the-art methods, is available at a monthly frequency and can be calculated in real-time, even when some indicators are not yet available for the most recent periods. The index is based on a large and broad set of monthly and quarterly indicators. As I show, for the case of Switzerland, it is important to base a business cycle index on a broad set of indicators instead of only a small subset. This result confirms the findings of a previous study on tracking short-term economic developments in Switzerland and is in contrast with the results for other countries.

Appendices

Appendix A: The Kalman Filter

Following Hamilton (1994), for our state space model

$$\begin{aligned} y_t&= H_t \xi _t + \epsilon _t \end{aligned}$$

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$$\begin{aligned} \xi _t&= F \xi _{t-1} + e_t \end{aligned}$$

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$$\begin{aligned} \text {with }&\begin{bmatrix} e_t \\ \epsilon _t \end{bmatrix} \sim N \begin{bmatrix} Q&0 \\ 0&R \end{bmatrix}, \end{aligned}$$

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the Kalman filter applied with the parameters $\theta = \begin{bmatrix} \Lambda ^*&\Phi&\Sigma _{vv}&\Sigma _{ww} \end{bmatrix}'$ provides a time series of filtered states ${\{\xi _{i|i}\}}_{i=1}^T$ and their covariance matrices ${\{P_{i|i}\}}_{i=1}^T$. The Kalman smoother provides a time series of the smooth correspondences, ${\{\xi _{i|T}\}}_{i=1}^T$ and ${\{P_{i|T}\}}_{i=1}^T$.

1.1 Details on the Kalman procedure

First we define:

• $\xi _{t|t-1} = E[\xi _{t}| {\{y_i\}}_{i=1}^{t-1}]$, the forecast for $\xi _t$ given data on y up to $t-1$

• $y_{t|t-1} = E[y_{t}| {\{y_i\}}_{i=1}^{t-1}]$, the forecast for $y_t$ given data on y up to $t-1$

• $P_{t|t-1} = Var(\xi _{t}| {\{y_i\}}_{i=1}^{t-1})$, the covariance matrix of $\xi _t$

• $\xi _{t|t} = E[\xi _{t}|{\{y_i\}}_{i=1}^t]$, the update for $\xi _t$ given all information up to t

• $P_{t|t} = Var(\xi _{t}| {y_i\}}_{i=1}^{t})$, the covariance of $\xi _t$ given all information up to t

Given our parameters $\theta$, the Kalman filter is then just an application of theorems on two variables (in our case $\xi$ and y) that are jointly normal distributed. The estimation proceeds as follows:

• Step 1, forecasting $\xi _t$: $\xi _{t|t-1} = F \xi _{t-1|t-1}$

• Step 2, variance of forecast for $\xi _t$: $P_{t|t-1} = F P_{t-1|t-1} F' + Q$

• Step 3, forecasting $y_t$: $y_{t|t-1} = H_t \xi _{t|t-1}$

• Step 4, variance of forecast for $y_t$: $S_t = H_t P_{t|t-1} H_t' + R$

• Step 5, forecast error of $y_t$: $z_t = y_t - y_{t|t-1}$

• Step 6, updating $\xi _t$: $\xi _{t|t} = \xi _{t|t-1} + P_{t|t-1} H'_t S_t^{-1} z_t$

• Step 7, variance of update for $\xi _t$: $P_{t|t} = P_{t|t-1} - P_{t|t-1} H'_t S_t^{-1} H_t P_{t|t-1}$

• Step 8, Kalman gain: $K_t = F P_{t|t-1} H'_t S_t^{-1}$

• Do the same for $\{\xi _{t+1}, P_{t+1}\}$, $\{\xi _{t+2}, P_{t+2}\}$, and so on.

The result is a time series of filtered states ${\{\xi _{i|i}\}}_{i=1}^T$ and their covariance matrices ${\{P_{i|i}\}}_{i=1}^T$. In the filtered observation vector, missing values have been filled by the Kalman filter.

1.1.1 Details on the Smoothing Procedure

From the Kalman filter, we obtain estimates of $\xi _{t|t-1}$ and $P_{t|t-1}$ (the forecasts, based on past information) as well as of $\xi _{t|t}$ and $P_{t|t}$ (the updates, based on past and current information) for each time period 1 to T. Starting with $\xi _{T|T}$ (the update for the last time period), we can go backwards to smooth our estimates by also incorporating future estimates and information, i.e., the estimation error for the next period given all information.

For $t=T$, the smoothed estimate is simply given by the filtered state, i.e.

$$\begin{aligned} \xi _{t|T} = \xi _{t|t} \end{aligned}$$

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and for $t<T$ by

$$\begin{aligned} \xi _{t|T} = \xi _{t|t} + P_{t|t} F' P^{-1}_{t+1|t} ( \xi _{t+1|T} - \xi _{t+1|t}), \end{aligned}$$

(34)

The result is a time series of smoothed states ${\{\xi _{i|T}\}}_{i=1}^T$. The mean squared error (MSE) of the smoothed states $\xi _{t|T}$ is given by

$$\begin{aligned} P_{t|T} = P_{t|t} + J_t (P_{t+1|T} - P_{t+1|t} ) J_t' \end{aligned}$$

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Appendix B

See Figs. 15, 16, 17, 18, 19, 20 and Table 3.

Fig. 15

The four factors of the business cycle index. Note: Shown are the four BCI factors

Fig. 16

Separate contributions to business cycle index by indicator categories. Note: Shown are the BCI contributions from the different indicator categories in deviations from mean

Fig. 17

Contributions to business cycle index by indicator types. Note: Shown are the BCI contributions from the different indicator types in deviations from mean

Fig. 18

Separate contributions to business cycle index by indicator types. Note: Shown are the BCI contributions from the different indicator types in deviations from mean

Table 3 Legend for the acronyms of the 30 most important indicators

Fig. 19

Availability of indicators in the data set. Note: Shown is the availability of all quarterly (top panel) and monthly (bottom panel) indicators in the data set. Green colors indicate that the indicator is available for the respective period, red colors indicate that the indicator is unavailable (color figure online)

Fig. 20

Comparison between real-time and replicated real-time data set. Note: Shown is the estimated business cycle index for the data vintage of 6th March 2012 based on real-time data (blue) and based on replicated real-time data (red) (color figure online)