21.4 IRREGULARLY OBSERVED AND UNOBSERVED DATA

21.4 IRREGULARLY OBSERVED AND UNOBSERVED DATA

Economic data are not always observed at equidistant points in time, and instead arrive at irregular times and may not even be observed on certain dates. Different methods have been developed to address this issue as we next describe.

21.4.1 Nowcasting

Nowcasting refers to “forecasting the present.” Specifically, it is the attempt to extract information about the present state of some variable or system of variables and so is distinct from traditional forecasting; see Giannone, Reichlin, and Small (2008). Nowcasting makes sense only if the present state is unknown—otherwise nowcasting would trivially amount to checking the current value of the variable of interest. Hence, while at first nowcasting may seem like a strange concept, it is easily understood in the context of models with one or more unobserved state variables. Suppose there is a single state variable that summarizes the state of the economy, e.g., the daily point in the business cycle. Suppose, however, that this state variable is unobservable, yet related to variables that are observable. Extracting the best estimate of this “summary” state variable could well be of interest to the prediction of many other variables that depend on the current state of the economy.

There are two key reasons for approaching the nowcasting problem as a system of state variables. First, as we have seen in our discussion of real-time data, even supposedly observed variables such as GDP are actually observed with measurement error. Second, because the economy is an interconnected system, information about underlying common factors can benefit from modeling the joint dynamics across multiple variables, many of which are affected by measurement error to varying degrees or simply not observed very frequently.

Many practical difficulties arise when constructing estimates of the current state of the economy or similar, unobserved, state variables at a relatively high frequency such as the daily horizon. Macroeconomic data such as GDP figures, monetary aggregates, consumption, unemployment figures, or housing starts, as well as financial data extracted from balance sheets and income statements, are published infrequently and sometimes at irregular intervals. Equivalently, the delay in the publication of macro variables differs across variables. Such nonsynchronous data releases give rise to what is often called “jagged edge” data (Banbura, Giannone, and Reichlin, 2011).

At any given point in time, a forecaster can use only such data as are available on that date and so needs to pay careful attention to which observables are in the information set. Let $Z_v=\{x_{i_t},t_i=1,\dots ,T_{iv},i=1,\dots ,n\}$ denote the information set containing the data available on day v, where $T_{iv}$ is the date of the most recent release of variable i .

Suppose an $(n\times 1)$ vector of monthly data $x_t=(x_{1t},x_{2t},...,x_{nt}{)}^{\prime }$ is available. Following Banbura, Giannone, and Reichlin (2011), assume that all variables have been transformed so that they are jointly stationary and lend themselves to a factor representation of the form

$$x_t=\mu +\Lambda F_t+{\epsilon }_t.\text{\hspace{1em}(21.12)}$$

As in chapter 10, $F_t$ comprises r unobserved factors, are factor loadings, and ${\epsilon }_t$ are idiosyncratic shocks. The factors have been demeaned, so $E\left[F_t\right]=0$ . We shall assume that the factors follow a first-order VAR process,

$$F_t=A{F}_{t-1}+u_t,u_t\sim \mathrm{ind}\mathrm{N}(0,Q).\text{\hspace{1em}(21.13)}$$

Finally, as in Banbura, Giannone, and Reichlin (2011), the idiosyncratic shocks in (21.12) are assumed to be driven by mutually uncorrelated AR(1) processes,

$${\epsilon }_t=\mathrm{diag}(\mathbf{\alpha }){\epsilon }_{t-1}+e_t,e_t\sim \mathrm{ind}\mathrm{N}(0,\mathrm{diag}({\sigma }^2)),\text{\hspace{1em}(21.14)}$$

so that $E[e_{it}{e}_{jt}]=0\mathrm{for}i\ne j$ . The terms diag(α) and $\mathrm{diag}({\sigma }^2)$ are both diagonal $(n\times n)$ matrices with values ${\alpha }_i$ and ${\sigma }_i^2$ , respectively, in the (i, i ) position on the main diagonals and 0s elsewhere.

The nowcasting approach can be illustrated by going from monthly to quarterly observations in a model for flow variables that are averaged across multiple periods. Denote monthly variables by a superscript M, $\Delta y_t^{\mathrm{M}}$ , and let $y_t=y_t^{\mathrm{M}}-y_{t-1}^{\mathrm{M}}$ be the monthly change—or growth rate if the variables are measured in logs. Suppose the monthly growth rates are unobserved but driven by the same factors as in (21.12). If the monthly data can be represented through a factor model with i.i.d. normal

shocks, we have

$$\begin{array}{rl}& {\mathbf{y}}_t^{\mathrm{M}}={\mathbf{\mu }}^{\mathrm{M}}+{\mathbf{\Lambda }}^{\mathrm{M}}{\mathbf{F}}_t^{\mathrm{M}}+{\epsilon }_t^{\mathrm{M}},\\ & {\epsilon }_t^{\mathrm{M}}={\alpha }^{\mathrm{M}}{\epsilon }_{t-1}^{\mathrm{M}}+u_t^{\mathrm{M}},u_t^{\mathrm{M}}\sim \mathrm{ind}\mathrm{N}(0,{\sigma }_{\mathrm{M}}^2).\end{array}\text{\hspace{1em}(21.15)}$$

Suppose we are interested in modeling quarterly variables such as the GDP and denote such variables by a superscript $\bar{\mathrm{Q}},y_t^{\mathrm{Q}}$ , so that $y_t^{\mathrm{Q}}=(y_t^{\mathrm{M}}+y_{t-1}^{\mathrm{M}}+y_{t-2}^{\mathrm{M}})$ The quarter-on-quarter rate of change, $\Delta y_t^{\mathrm{Q}}\equiv y_t^{\mathrm{Q}}-y_{t-3}^{\mathrm{Q}}.$ , can be approximated by Banbura, Giannone, and Reichlin (2011),

$$\Delta y_t^{\mathrm{Q}}\approx y_t^{\mathrm{M}}+2y_{t-1}^{\mathrm{M}}+3y_{t-2}^{\mathrm{M}}+2y_{t-3}^{\mathrm{M}}+y_{t-4}^{\mathrm{M}},\text{\hspace{1em}(21.16)}$$

where t now increases in increments of 3, i.e., $t=3,6,9,12$ . The term $\Delta y_t$ is a weighted average of monthly growth rates. Notice how the weights peak in the middle and have a tent-shaped pattern. This is a typical pattern when we go from a higher frequency (monthly) to a lower frequency (quarterly) and sum across growth rates.

Defining ${\bar{\mathbf{x}}}_t=({\dot{x}}_t^{\prime },\Delta y_t^{\mathrm{Q}}{)}^{\prime }$ , Banbura, Giannone, and Reichlin (2011) show that (21.12)–(21.16) can be written as a state-space model:

$$\begin{array}{l}{\bar{x}}_t=\bar{\mu }+Z(\theta ){\bar{\xi }}_t,\\ {\bar{\xi }}_t=T(\theta ){\bar{\xi }}_{t-1}+{\eta }_t,{\eta }_t\sim \mathrm{ind}\mathrm{N}(0,{\Sigma }_{\eta }(\theta )),\end{array}$$

where the state vector ${\bar{\xi }}_t$ and the parameters θ are given by

$$\begin{array}{rl}& {\bar{\xi }}_t=(F_t^{\mathrm{M}},F_{t-1}^{\mathrm{M}},F_{t-2}^{\mathrm{M}},F_{t-3}^{\mathrm{M}},F_{t-4}^{\mathrm{M}},{\epsilon }_t^{\prime },{\epsilon }_t^{\mathrm{M}},{\epsilon }_{t-1}^{\mathrm{M}},{\epsilon }_{t-2}^{\mathrm{M}},{\epsilon }_{t-3}^{\mathrm{M}},{\epsilon }_{t-4}^{\mathrm{M}}{)}^{\prime },\\ & \theta =\{\mu ,{\mu }^{\mathrm{M}},\Lambda ,{\Lambda }^{\mathrm{M}},A,\mathrm{diag}(\alpha ),{\alpha }^{\mathrm{M}},\mathrm{diag}({\sigma }^2),{\sigma }_{\mathrm{M}}^2\}.\end{array}$$

Note that four lags are needed due to the weights in (21.16). For the single-factor $(r=1)$ univariate case, the model gives rise to the following measurement equations:

$$\left(\begin{array}{l}{x}_t\\ \Delta y_t^{(a)}\\ \Delta y_t^{(a)}\end{array}\right)=\left(\begin{array}{l}\mu \\ g_{\mu }u\end{array}\right)+\left(\begin{array}{lllllllcc}\Lambda & 0& 0& 0& I_n& 0& 0& 0& \\ {\Lambda }^M& 2{\Lambda }^M& 3{\Lambda }^M& 2{\Lambda }^M& {\Lambda }^M& 0& 1& 2& 1\\ {\Lambda }^M& 2{\Lambda }^M& 2{\Lambda }^M& 2{\Lambda }^M& 2& 1& 2& 1& \end{array}\right)\left[\begin{array}{l}{F}_n^{(a)}\\ F_{n-1}^{(a)}\\ F_{n-2}^{(a)}\\ F_{n-3}^{(a)}\end{array}\right]$$

Similarly, the state equations take the form

( \begin{array} {l} { F _ { \mathrm{H}} ^ { \mathrm{H}} } \\ { F _ { \mathrm{H}} ^ { \mathrm{H}} } \\ { F _ { \mathrm{H}} ^ { \mathrm{M}} } \\ { F _ { \mathrm{L}} ^ { \mathrm{M}} } \\ { F _ { \mathrm{L}} ^ { \mathrm{M}} } \\ { F _ { \mathrm{H}} ^ { \mathrm{H}} } \\ { F _ { \mathrm{H}} ^ { \mathrm{H}} } \\ { E _ { \mathrm{H}} ^ { \mathrm{H}} } \\ { \varepsilon _ { \mathrm{H}} ^ { \mathrm{H}} } \\ { \varepsilon _ { \mathrm{H}} ^ { \mathrm{H}} } \\ { \varepsilon _ { \mathrm{H}} ^ { \mathrm{H}} } \\ { \varepsilon _ { \mathrm{H}} ^ { \mathrm{H}} } \\ { \varepsilon _ { \mathrm{H}} ^ { \mathrm{H}} } \\ { \varepsilon _ { \mathrm{H}} ^ { \mathrm{H}} } \\ { \varepsilon _ { \mathrm{H}} ^ { \mathrm{H}} } \\ { \varepsilon _ { \mathrm{H}} ^ { \mathrm{H}} } \\ { \varepsilon _ { \mathrm{H}} ^ { \mathrm{H}} } \\ { \varepsilon _ { \mathrm{H}} ^ { \mathrm{H}} } \\ { \varepsilon _ { \mathrm{H}} ^ { \mathrm{H}} } \\ { \varepsilon _ { \mathrm{H}} ^ { \mathrm{H}} } \end{array} ) = ( \begin{array} {llllllll} { A _ { \mathrm{~0}} \ \ \mathrm{~0} \ \mathrm{~0} \ \ \mathrm{~0} \ \ \mathrm{~0} \ \ \mathrm{~0} \ } & { \mathrm{~0} \ } & { \mathrm{~0} \ \ \mathrm{~0} \ \ \mathrm{~0} \ \ \mathrm{~0} \ } \\ { I _ { \mathrm{~r}} \ \mathrm{~0} \ \ \mathrm{~0} \ \ \mathrm{~0} \ \ \mathrm{~0} \ \ \mathrm{~0} \ \ \mathrm{~0} \ \ \mathrm{~0} \ \ 0 \ \mathrm{~0} \ \mathrm{~0} \ } \\ 0 \ \mathrm{~0} \ \mathrm{~0} \ \ \mathrm \end{array}

Using these equations, the model can be estimated by means of a Kalman filter. In turn, the extracted states can be used to generate forecasts of monthly and quarterly growth rates. See Banbura, Giannone, and Reichlin (2011) for additional details and further discussion.