Chow-Lin, Fernandez and Litterman

Eurostat (2018) recommends the use of regression-based models for the purpose of temporal disaggregation. Among them, we retrieve the Chow-Lin method and its variants Fernandez and Litterman.

Let Y_T, T = 1, ..., m, and x_t, t = 1, ..., n, be, respectively the observed low frequency benchmark and the high-frequency indicator of an unknown high frequency variable y_t. Chow-Lin, Fernandez and Litterman can be all expressed with the same equation, but with different models for the error term: y_t = x_tβ + u_t where

u_t = ϕu_t − 1 + ϵ_t, with |ϕ| < 1 (Chow-Lin),

u_t = u_t − 1 + ϵ_t (Fernandez),

u_t = u_t − 1 + ϕ(Δu_t − 1) + ϵ_t, with |ϕ| < 1 (Litterman)

While x_t is observed in high frequency, y_t is only observed in low frequency, and therefore the number of effective observations to estimate the parameters are the number of observations in the low-frequency benchmark.

Regression-based methods can be called with the temporaldisaggregation() function.

# Example: Use of Fernandez variant to disaggregate annual value added in construction sector using a quarterly indicator
Y <- ts(qna_data$B1G_Y_data[, "B1G_FF"], frequency=1, start=c(2009, 1))
x <- ts(qna_data$TURN_Q_data[, "TURN_INDEX_FF"], frequency=4, start=c(2009, 1))
td_fern <- rjd3bench::temporaldisaggregation(Y, indicators=x, model = "Rw")

y_fern <- td_fern$estimation$disagg # the disaggregated series
summary(td_fern)
plot(td_fern)

The output of the temporaldisaggregation() function contains the most important information about the regression including the estimates of model coefficients and their covariance matrix, the decomposition of the disaggregated series and information about the residuals. The print(), summary() and plot() functions can also be applied on the output object. The plot() function displays the decomposition of the disaggregated series between regression and smoothing effect.

Model-based Denton

Denton method and variants are usually expressed in mathematical terms as a constrained minimization problem. For example, the widely used Denton proportional first difference (PFD) method is usually expressed as follows: $$ min_{y_t}\sum^n_{t=2}\biggl[\frac{y_t}{x_t}-\frac{y_{t-1}}{x_{t-1}}\biggr]^2 $$ subject to the temporal constraint (flow variables) ∑_ty_t = Y_T where y_t is the value of the estimate of the high frequency series at period t, x_t is the value of the high frequency indicator at period t and Y_T is the value of the low frequency series (i.e. the benchmark series) at period T.

Equivalently, the Denton PFD method can also be expressed as a statistical model considering the following state space representation

$$ \begin{aligned} y_t &= \beta_t x_t \\ \beta_{t+1} &= \beta_t + \varepsilon_t \qquad \varepsilon_t \sim {\sf NID}(0, \sigma^2_{\varepsilon}) \end{aligned} $$

where the temporal constraints are taken care of by considering a cumulated series y_t^c instead of the original series y_t. Hence, the last high frequency period (for example, the last quarter of the year) is observed and corresponds to the value of the benchmark. The value of the other periods are initially defined as missing and estimated by maximum likelihood.

This alternative representation of Denton PFD method is interesting as it allows more flexibility. We might now include outliers - namely, level shift(s) in the Benchmark to Indicator ratio - that could otherwise induce undesirable wave effects. Outliers and their intensity are defined by changing the value of the innovation variances. There is also the possibility to freeze the disaggregated series at some specific period(s) or prior a certain date by fixing the high-frequency BI ratio(s). Following the principle of movement preservation inherent to Denton, the model-based Denton PFD method constitutes an interesting alternative for both temporal disaggregation and benchmarking. Here is a link to a presentation on the subject which include some comparison with the regression-based methods for temporal disaggregation.

The model-base Denton method can be applied with the denton_modelbased() function.

# Example: Use of model-based Denton for temporal disaggregation
Y <- ts(qna_data$B1G_Y_data[, "B1G_FF"], frequency=1, start=c(2009, 1))
x <- ts(qna_data$TURN_Q_data[, "TURN_INDEX_FF"], frequency=4, start=c(2009, 1))
td_mbd <- rjd3bench::denton_modelbased(Y, x, outliers = list("2020-01-01"=100, "2020-04-01"=100))

y_mbd <- td_mbd$estimation$disagg
plot(td_mbd)

The output of the denton_modelbased() function contains information about the disaggregated series and the BI ratio as well as their respecting errors making it possible to construct confidence intervals. The print(), summary() and plot() functions can also be applied on the output object.The plot() function displays the disaggregated series and the BI ratio together with their respective 95% confidence interval.

Autoregressive Distributed Lag (ADL) Models

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Denton

Denton methods relies on the principle of movement preservation. There exist several variants corresponding to different definitions of movement preservation: additive first difference (AFD), proportional first difference (PFD), additive second difference (ASD), proportional second difference (PSD).

The most widely used is the Denton PFD variant. Let Y_T, T = 1, ..., m, and x_t, t = 1, ..., n, be, respectively the temporal benchmarks and the high-frequency preliminary values of an unknown target variable y_t. The objective function of the Denton PFD method is as follows (considering the small modification suggested by Cholette to deal with the starting conditions of the problem): $$ min_{y_t}\sum^n_{t=2}\biggl[\frac{y_t}{x_t}-\frac{y_{t-1}}{x_{t-1}}\biggr]^2 $$ This objective function is minimized subject to the temporal aggregation constraints ∑_tϵT = Y_T, T = 1, ..., m (flows variables). In other words, the benchmarked series is estimated in such a way that the “Benchmark-to-Indicator” ratio $\frac{y_t}{x_t}$ remains as smooth as possible, which is often of key interest in benchmarking.

In the literature (see for example Di Fonzo and Marini, 2011), Denton PFD is generally considered as a good approximation of the GRP method, meaning that it preserves the period-to-period growth rates of the preliminary series. It is also argued that in many applications, Denton PFD is more appropriate than GRP method as it deals with a linear problem which is computationally easier, and does not suffer from the issues related to time irreversibility and singular objective function when y_t approaches 0 (see Daalmans et al, 2018).

Denton methods can be called with the denton() function.

# Example: use Denton method for benchmarking
Y <- ts(qna_data$B1G_Y_data[, "B1G_HH"], frequency=1, start=c(2009, 1))

y_den0 <- rjd3bench::denton(t=Y, nfreq=4) # denton PFD without high frequency series

x <- y_den0 + rnorm(n=length(y_den0), mean=0, sd=10)
y_den1 <- rjd3bench::denton(s=x, t=Y) # denton PFD (= the default)
y_den2 <- rjd3bench::denton(s=x, t=Y, d=2, mul=FALSE) # denton ASD

The denton() function returns the benchmarked high frequency series.

Growth rate preservation (GRP)

GRP explicitly preserves the period-to-period growth rates of the preliminary series.

Let Y_T, T = 1, ..., m, and x_t, t = 1, ..., n, be, respectively the temporal benchmarks and the high-frequency preliminary values of an unknown target variable y_t. Cauley and Trager(1981) consider the following objective function:

$$ f(x) = \sum_{t=2}^{n}\left(\frac{y_t}{y_{t-1}} - \frac{x_t}{x_{t-1}}\right)^2 $$ and look for values y_t^*, t = 1, ..., n, which minimize it subject to the temporal aggregation constraints ∑_tϵT = Y_T, T = 1, ..., m (flows variables). In other words, the benchmarked series is estimated in such a way that its temporal dynamics; as expressed by the growth rates $\frac{y_t^*}{y_{t-1}^*}$, t = 2, ..., n, be “as close as possible” to the temporal dynamics of the preliminary series, where the “distance” from the preliminary growth rates $\frac{x_t}{x_{t-1}}$ is given by the sum of the squared differences. (Di Fonzo, Marini, 2011)

The objective function considered by Cauley and Trager is a natural measure of the movement of a time series and as one would expect, it is usually slightly better than the Denton PFD method at preserving the movement of the series (Di Fonzo, Marini, 2011). However, unlike the Denton PFD method which deals with a linear problem, GRP solves a more difficult nonlinear problem. Furthermore, the GRP method suffers from a couple of drawbacks, which are time irreversibility and potential singularities in the objective function when y_t − 1 approaches to 0, which could lead to undesirable results (see Daalmans et al, 2018).

The GRP method, corresponding to the method of Cauley and Trager, using the solution proposed by Di Fonzo and Marini (2011), can be called with the grp() function.

# Example: use GRP method for benchmarking
Y <- ts(qna_data$B1G_Y_data[, "B1G_HH"], frequency=1, start=c(2009, 1))
y_den0 <- rjd3bench::denton(t=Y, nfreq=4)
x <- y_den0 + rnorm(n=length(y_den0), mean=0, sd=10)

y_grp <- rjd3bench::grp(s=x, t=Y)

The grp() function returns the high frequency series benchmarked with the GRP method.

Cubic splines

Cubic splines are piecewise cubic functions that are linked together in a way to guarantee smoothness at data points. Additivity constraints are added for benchmarking purpose and sub-period estimates are derived from each spline. When a sub-period indicator (or disaggregated series) is used, cubic splines are no longer drawn based on the low frequency data but the Benchmark-to-Indicator (BI ratio) is the one being smoothed. Sub-period estimates are then simply the product between the smoothed high frequency BI ratio and the indicator.

The method can be called through the cubicspline() function. Here are a few examples on how to use it:

y_cs1<-rjd3bench::cubicspline(t=Y, nfreq=4) # example of cubic spline without high frequency series (smoothing)

x<-y_cs1+rnorm(n=length(y_cs1), mean=0, sd=10)
y_cs2<-rjd3bench::cubicspline(s=x, t=Y) # example of cubic spline with a high frequency series to benchmark

The cubicspline() function returns the high frequency series benchmarked with cubic spline method.

Cholette method

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Multivariate Cholette

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