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21.4.2 Mixed Data Sampling Methods

Can we improve on the forecasts of, say, monthly or quarterly variables, by utilizing data observed more often such as daily stock prices and interest rates? This is an important question since variables such as stock prices and interest rates are published on a daily basis, while payroll figures are published weekly, industrial production gets published monthly, and GDP figures are published quarterly and are subject to revisions. To make full use of all information, forecasters effectively have to use mixed frequency data. A string of papers summarized in Andreou, Ghysels, and Kourtellos (2011) have developed the so-called MIDAS—mixed data sampling— approach to estimation and forecasting that addresses this situation.6

Suppose we are interested in predicting a quarterly variable, $Y_{t + 1}^{Q}$ , using daily observations on a predictor, $X_{N_{D} - j, t}^{D},$ where $N_{D}$ is the number of days in the quarter, $j = 0$ refers to the last day of quarter $t, X_{N_{D}, t},$ while $j = N_{D} - 1$ refers to the first day of quarter $t, X_{1, t}$ . Note the double subscript X: the first subscript refers to days and counts backwards, while the second subscript refers to the quarter.

One option is to use only the average value of the daily variable,

$\overset{ˉ}{X}_{t}^{Q} = (X_{N_{D}, t}^{D} + X_{N_{D} - 1, t}^{D} + \cdot \cdot \cdot + X_{1, t}^{D}) / N_{D},$

as conditioning information in the prediction model. This measure puts the same weight on older and more recent daily observations and may not be the best approach because older information could be less relevant than more recent information.

To deal with this deficiency, another option would be to simply regress the quarterly outcome variable on all daily variables observed during the most recent and perhaps previous quarters, ${X_{N_{D}, t}^{D}, \dots, X_{1, t}^{D}, X_{N_{D, t - 1}}^{D}, \dots}$ . However, this introduces many new parameters and is likely to lead to large parameter estimation errors.

As an alternative solution to these two extremes, Ghysels, Sinko, and Valkanov (2007) propose to use data-driven weighting (or aggregation) schemes that in a parsimonious manner allow more weight to be placed on recent daily observations without discarding old data points by applying lag polynomials to high-frequency data. They consider an exponential Almon lag and a beta lag polynomial, both of which contain two unknown parameters $(θ_{1}, θ_{2})$

The weight of the exponential Almon lag polynomial at lag j takes the form

$w_{j} (θ_{1}, θ_{2}) = \frac{exp ( θ _{1} j + θ _{2} j ^{2} )}{\sum _{j = 1}^{m} exp ( θ _{1} j + θ _{2} j ^{2} )}, 1 \leq j \leq m, (21.17)$

where m is some truncation point. The weights of the beta lag are

$w_{j} (θ_{1}, θ_{2}) = \frac{g ( j ; θ _{1} , θ _{2} )}{\sum _{j = 1}^{m} g ( j ; θ _{1} , θ _{2} )}, (21.18)$

$g (j; θ_{1}, θ_{2}) = j^{θ_{1} - 1} (1 - j)^{θ_{2} - 1} \frac{Γ ( θ _{1} + θ _{2} )}{Γ ( θ _{1} ) Γ ( θ _{2} )},$

$Γ (θ_{1}) = \int_{0}^{\infty} e^{- x} x^{θ_{1} - 1} d x .$

Given a set of estimates of $θ_{1}$ and $θ_{2} .$ , the weights on the individual daily observations can be computed and the quarterly variable can be projected on the weighted average of the daily observations:

$Y_{t + 1}^{Q} = μ + β j = 0 \sum N_{D} - 1 w_{j} (θ_{1}, θ_{2}) X_{N_{D} - j, t}^{D} + u_{t + 1} . (21.19)$

For simplicity we assumed here that only the daily data from the previous quarter are used in the forecast, but daily observations from prior periods can of course also be used with the weights (and the cutoff point, m) adjusted accordingly. For example, Ghysels, Santa-Clara, and Valkanov (2005) consider a cutoff of 250 data points.

The MIDAS approach is quite flexible and encompasses equal weights, for which (21.19) simplifies to

$Y_{t + 1}^{Q} = μ + β \overset{ˉ}{X}_{t}^{Q} + u_{t + 1} .$

This case arises as a special case of the exponential Almon lag in (21.17) with $θ_{1} = θ_{2} = 0$

It is clear from this description that once the parameters $(θ_{1}, θ_{2})$ have been determined, the prediction step in (21.19) is trivial and involves only linear projection, making the approach easy to use.

练习题

What is the main purpose of using mixed frequency data in forecasting?

A. To simplify the forecasting model

B. To make full use of all available information

C. To reduce the number of parameters in the model

D. To increase the frequency of data collection

When predicting a quarterly variable $Y_{t + 1}^{Q}$ using daily observations, what does the subscript $N_{D} - j, t$ in $X_{N_{D} - j, t}^{D}$ represent?

A. The number of days in the year

B. The number of days in the quarter, counting backwards from the last day

C. The total number of observations in the dataset

D. The number of quarters in the year

What is a potential drawback of using only the average value of daily variables as conditioning information in a prediction model?

A. It puts more weight on recent observations

B. It puts the same weight on older and more recent daily observations

C. It requires fewer parameters to estimate

D. It simplifies the model structure

What are the potential issues with regressing the quarterly outcome variable on all daily variables observed during the most recent and previous quarters?

A. It introduces many new parameters

B. It may lead to large parameter estimation errors

C. It simplifies the model structure

D. It requires less computational power

E. It improves the accuracy of forecasts

What are the advantages of using data-driven weighting schemes in mixed frequency data forecasting?

A. They allow more weight to be placed on recent daily observations

B. They discard old data points

C. They are parsimonious

D. They apply lag polynomials to high-frequency data

E. They reduce the need for parameter estimation

The exponential Almon lag polynomial puts more weight on older observations as the lag $j$ increases.

The beta lag polynomial includes a term $Γ (θ_{1} + θ_{2})$ in its weight formula.

The weight of the exponential Almon lag polynomial at lag $j$ is given by $w_{j} (θ_{1}, θ_{2}) = \frac{e x p ( θ _{1} j + θ _{2} j ^{2} )}{\sum _{j = 1}^{m} e x p ( θ _{1} j + θ _{2} j ^{2} )}$ , where $m$ is the ___.

The MIDAS approach simplifies to using equal weights when $θ_{1} = θ_{2} =$ ___.

Explain why the MIDAS approach is considered easy to use once the parameters $(θ_{1}, θ_{2})$ have been determined.

Which of the following are true about the beta lag polynomial?

A. It includes a term

j^{θ_{1} - 1}

B. It includes a term

(1 - j)^{θ_{2} - 1}

C. It uses the gamma function

Γ (θ_{1} + θ_{2})

D. It is only applicable to quarterly data

E. It simplifies to equal weights when

θ_{1} = θ_{2} = 0

How does the MIDAS approach handle the issue of older information being less relevant than more recent information?

When predicting a quarterly variable $Y_{t + 1}^{Q}$ using daily observations $X_{N_{D} - j, t}^{D}$ , which of the following approaches is most likely to lead to large parameter estimation errors?

A. Using the average value of the daily variable

\overset{ˉ}{X}_{t}^{Q}

as conditioning information

B. Regressing the quarterly outcome variable on all daily variables observed during the most recent and previous quarters

C. Applying data-driven weighting schemes with exponential Almon lag polynomials

D. Using the beta lag polynomial for weighting daily observations

Which of the following statements about the MIDAS approach are correct?

A. The MIDAS approach can use equal weights as a special case

B. The MIDAS approach is only applicable to exponential Almon lag polynomials

C. The MIDAS approach simplifies to a linear projection once parameters are determined

D. The MIDAS approach requires regressing on all daily variables

E. The MIDAS approach can use beta lag polynomials

In the MIDAS approach, the projection of the quarterly variable $Y_{t + 1}^{Q}$ on the weighted average of daily observations is given by $Y_{t + 1}^{Q} = μ + β \sum_{j = 0}^{N_{D} - 1} w_{j} (θ_{1}, θ_{2}) X_{N_{D} - j, t}^{D} + u_{t + 1}$ , where $w_{j} (θ_{1}, θ_{2})$ is determined by either the _ or the _ lag polynomial.

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21.4.2 Mixed Data Sampling Methods