Temporal Disaggregation of Time Series

Perform temporal disaggregation or interpolation of low frequency to high frequency time series. td can be used with objects of class "ts", with numeric vectors or with any ts-boxable time series object.

Usage

td(
  formula,
  conversion = "sum",
  to = "quarterly",
  method = "chow-lin-maxlog",
  truncated.rho = 0,
  fixed.rho = 0.5,
  criterion = "proportional",
  h = 1,
  start = NULL,
  end = NULL,
  ...
)

Arguments

formula: an object of class "formula": a symbolic description of the the temporal disaggregation model. The details of model specification are given under 'Details'.
conversion: type of conversion: "sum", "mean" (or: "average"), "first" or "last".
to: high-frequency destination frequency as a character string ("quarter" (or "quarterly"), "month" (or "monthly"), "day", "hour", "minute", "second", or "year") or as a scalar (e.g. 2, 4, 7, 12). Required if no right hand side indicator series is provided. The tsbox package must be installed to deal with frequencies other than monthly or quarterly. If the input series are numeric, to is a scalar indicating the frequency ratio.
method: method of temporal disaggregation: "chow-lin-maxlog", "chow-lin-minrss-ecotrim", "chow-lin-minrss-quilis", "chow-lin-fixed", "dynamic-maxlog" (experimental), "dynamic-minrss" (experimental), "dynamic-fixed" (experimental), "fernandez", "litterman-maxlog", "litterman-minrss", "litterman-fixed", "denton-cholette", "denton", "fast", "uniform" or "ols". See 'Details'.
truncated.rho: lower bound for the autoregressive parameter \(\rho\). If set to 0 (default), no negative values are allowed. If set to -1, truncation is disabled.
fixed.rho: set a predefined autoregressive parameter \(\rho\). Only works with the methods "chow-lin-fixed" and "litterman-fixed".
criterion: minimzation criterion for Denton methods: "proportional" or "additive". See 'Details'.
h: degree of differencing for Denton methods. See 'Details'.
start: (optional) start date. Similar to pre-processing the input series with window().
end: (optional) end date. Similar to pre-processing the input series with window().
...: additional arguments to be passed to the low level subfunctions.

Value

td returns an object of class "td".

The function predict() computes the interpolated high frequency series. If the high-frequency indicator series are longer than the low-frequency series, the resulting series will be extrapolated. The function coefficients extracts the coefficients. The function residuals extracts the low frequency residuals. The function summary() prints a summary of the estimation.

An object of class "td" is a list containing the following components:

values: disaggregated or interpolated (and extrapolated) high frequency series
fitted.values: low frequency fitted values of the regression; low frequency indicator for the Denton methods.
p: preliminary high frequency series
residuals: low-frequency residuals
rho: autoregressive parameter, \(\rho\)
truncated: logical, whether \(\rho\) has been truncated
coefficients: a named vector of coefficients
se: standard errors of the coefficients
s_2: ML-estimator of the variance of the high-frequency residuals
s_2_gls: GLS-estimator of the variance of the high-frequency residuals
tss: weighted (low frequency) total sum of squares
rss: weighted (low frequency) residual sum of squares
r.squared: R squared
adj.r.squared: adjusted R squared
logl: log-likelihood
aic: Akaike information criterion
bic: Schwarz information criterion
rank: number of right hand variables (including intercept)
df: degrees of freedom
method: method of temporal disaggregation
call: function call
name: name of the low frequency variable
fr: the ratio of high to low-frequency series
conversion: type of temporal conversion
actual: actual values of the low frequeny series
model: a matrix containing the indicators (and a constant if present)
criterion: minimization criterion in Denton methods
h: order of differencing in Denton methods

Details

td is used to disaggregate or interpolate a low frequency to a higher frequency time series, while either the sum, the average, the first or the last value of the resulting high-frequency series is consistent with the low frequency series. Disaggregation can be performed with or without the help of one or more right hand side indicator series. It can deal with both with a regular disaggregation setting (e.g. quarters to months) but also with an irregular disaggregation setting (e.g. months to days), where it respects the the different lengths of the months.

If the high-frequency indicator(s) cover(s) a longer time span than the low-frequency series, an extrapolation or retropolation (Wei, 1994, p. 138) is performed, using the same model as for interpolation.

The selection of a temporal disaggregation model is similar to the selection of a linear regression model. Thus, td closely mirrors the working of the lm() function. The left hand side of the formula() denotes the low-frequency series, the right hand side the indicators. If no indicator is specified, the right hand side must be set equal to 1 (see examples). Unlike lm, td handles ts() and mts time-series objects, as a typical application involves the use of these objects. Alternatively, If used with basic vectors, the to argument specifies the ratio between the high and the low frequency series.

For the generalized least squares (GLS) methods "chow-lin-maxlog", "chow-lin-minrss-ecotrim", "chow-lin-minrss-quilis", "litterman-maxlog" and "litterman-minrss", an autoregressive parameter \(\rho\) is estimated. Default (and recommended) method is chow-lin-maxlog. With truncated.rho = 0 (default), it produces good results for a wide range of applications.

There are two variants of the chow-lin-minrss approach that lead to different results: Ecotrim by Barcellan (2003) uses a correlation matrix instead of the variance covariance matrix (implemented in "chow-lin-minrss-ecotrim"), the Matlab library by Quilis (2009) multiplies the correlation matrix with \(1/(1-\rho^2)\) (implemented in "chow-lin-minrss-quilis").

The methods "dynamic-maxlog", "dynamic-minrss" and "dynamic-fixed" are dynamic extensions of Chow-Lin (Santos Silva and Cardoso, 2001). If the autoregressive parameter \(\rho\) is equal to 0, no truncation remainder is added.

The Denton methods "denton" and "denton-cholette" can be specified with one or without an indicator. The parameter h can be set equal to 0, 1, or 2. Depending on the value, the denton procedure minimizes the sum of squares of the deviations between the levels (0), the first differences (1) or the second differences (2) of the indicator and the resulting series. Additionally, criterion can be set equal to "proportional" or "additive", depending on whether the proportional or the absolute deviations should be considered for minimzation. "denton-cholette" removes the transient movement of the original "denton" method at the beginning of the resulting series. "fast" is a shortcut for "chow-lin-fixed" with fixed.rho = 0.99999. It returns approximately the same results as "denton-cholette" with h = 1, but is much faster.

"uniform" is a special case of the "denton" approach, with h equals 0 and criterion equals "additive". It distributes the residuals uniformly. If no indicator is used, this leads to a step-shaped series.

"ols" performs an ordinary least squares regression (OLS) and distributes the residuals uniformly. It is especially useful for comparing the estimators of GLS and OLS regressions.

References

Chow, G. C., & Lin, A. L. (1971). Best linear unbiased interpolation, distribution, and extrapolation of time series by related series. The review of Economics and Statistics, 372-375.

Denton, F. T. (1971). Adjustment of monthly or quarterly series to annual totals: an approach based on quadratic minimization. Journal of the American Statistical Association, 66(333), 99-102.

Santos Silva, J. M. C. & Cardoso, F. N. (2001). The Chow-Lin method using dynamic models. Economomic Modelling, 18, 269-280.

Wei, W. W. S. (1994). Time series analysis. Addison-Wesley publ.

Sax, C. und Steiner, P. (2013). Temporal Disaggregation of Time Series. The R Journal, 5(2), 80-88. doi:10.32614/RJ-2013-028

Examples

data(tempdisagg)

# one indicator, no intercept
mod1 <- td(sales.a ~ 0 + exports.q)
summary(mod1)  # summary statistics
#> 
#> Call:
#> td(formula = sales.a ~ 0 + exports.q)
#> 
#> Residuals:
#>    Min     1Q Median     3Q    Max 
#> -86.57  29.00  32.27  36.81  59.55 
#> 
#> Coefficients:
#>            Estimate Std. Error t value Pr(>|t|)    
#> exports.q 0.0141601  0.0003428   41.31   <2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> 'chow-lin-maxlog' disaggregation with 'sum' conversion
#> 36 low-freq. obs. converted to 158 high-freq. obs.
#> Adjusted R-squared: 0.9362	AR1-Parameter: 0.862
plot(mod1)  # residual plot of regression

plot(predict(mod1))


# interpolated quarterly series

# temporally aggregated series is equal to the annual value
all.equal(window(
  ta(predict(mod1), conversion = "sum", to = "annual"),
  start = 1975), sales.a)
#> [1] TRUE

# several indicators, including an intercept
mod2 <- td(sales.a ~ imports.q + exports.q)

# no indicator (Denton-Cholette)
mod3 <- td(sales.a ~ 1, to = "quarterly", method = "denton-cholette")

# no indicator (uniform)
mod4 <- td(sales.a ~ 1, to = "quarterly", method = "uniform")

# Dynamic Chow-Lin (Santos Silva and Cardoso, 2001)
# (no truncation parameter added, because rho = 0)
mod5 <- td(sales.a ~ exports.q, method = "dynamic-maxlog")

# Example from Denton (1971), see references.
d.q <- ts(rep(c(50, 100, 150, 100), 5), frequency = 4)
d.a <- ts(c(500, 400, 300, 400, 500))

a1 <- predict(td(d.a ~ 0 + d.q, method = "denton",
                 criterion = "additive", h = 0))
#> 'denton-cholette' removes the transient movement at the beginning of the series and is preferable to the original 'denton' method in most cases.
a2 <- predict(td(d.a ~ 0 + d.q, method = "denton",
                 criterion = "additive", h = 1))
#> 'denton-cholette' removes the transient movement at the beginning of the series and is preferable to the original 'denton' method in most cases.
a3 <- predict(td(d.a ~ 0 + d.q, method = "denton",
                 criterion = "additive", h = 2))
#> 'denton-cholette' removes the transient movement at the beginning of the series and is preferable to the original 'denton' method in most cases.
a4 <- predict(td(d.a ~ 0 + d.q, method = "denton",
                 criterion = "additive", h = 3))
#> 'denton-cholette' removes the transient movement at the beginning of the series and is preferable to the original 'denton' method in most cases.

p1 <- predict(td(d.a ~ 0 + d.q, method = "denton",
                 criterion = "proportional", h = 0))
#> 'denton-cholette' removes the transient movement at the beginning of the series and is preferable to the original 'denton' method in most cases.
p2 <- predict(td(d.a ~ 0 + d.q, method = "denton",
                 criterion = "proportional", h = 1))
#> 'denton-cholette' removes the transient movement at the beginning of the series and is preferable to the original 'denton' method in most cases.
p3 <- predict(td(d.a ~ 0 + d.q, method = "denton",
                 criterion = "proportional", h = 2))
#> 'denton-cholette' removes the transient movement at the beginning of the series and is preferable to the original 'denton' method in most cases.
p4 <- predict(td(d.a ~ 0 + d.q, method = "denton",
                 criterion = "proportional", h = 3))
#> 'denton-cholette' removes the transient movement at the beginning of the series and is preferable to the original 'denton' method in most cases.

# Table in Denton (1971), page 101:
round(cbind(d.q, a1, a2, a3, a4, p1, p2, p3, p4))
#>      d.q  a1  a2  a3  a4  p1  p2  p3  p4
#> 1 Q1  50  75  67  62  59  56  57  55  54
#> 1 Q2 100 125 127 125 123 122 124 122 120
#> 1 Q3 150 175 180 182 184 200 194 194 195
#> 1 Q4 100 125 126 130 134 122 125 129 132
#> 2 Q1  50  50  65  70  74  50  58  61  62
#> 2 Q2 100 100 105 106 107 100 107 109 111
#> 2 Q3 150 150 145 142 141 150 146 145 144
#> 2 Q4 100 100  85  81  78 100  89  85  83
#> 3 Q1  50  25  27  24  22  44  40  39  38
#> 3 Q2 100  75  73  72  71  78  74  73  72
#> 3 Q3 150 125 123 124 125 100 109 110 111
#> 3 Q4 100  75  78  80  81  78  77  78  79
#> 4 Q1  50  50  37  38  39  50  43  43  43
#> 4 Q2 100 100  96  96  96 100  94  94  94
#> 4 Q3 150 150 154 155 154 150 153 154 153
#> 4 Q4 100 100 112 112 111 100 110 110 110
#> 5 Q1  50  75  69  68  67  56  58  58  58
#> 5 Q2 100 125 124 123 123 122 123 121 122
#> 5 Q3 150 175 178 178 178 200 190 189 190
#> 5 Q4 100 125 129 132 132 122 129 131 130

if (FALSE) { # \dontrun{

# Using altvernative time series classes (see https://docs.ropensci.org/tsbox/)
library(tsbox)
sales.a.xts <- ts_xts(window(sales.a, start = 2000))
exports.q.xts <- ts_xts(window(exports.q, start = 2000))
mod1b <- td(sales.a.xts ~ 0 + exports.q.xts)
predict(mod1b)  # class 'xts'

# non-standard frequencies: decades to years
predict(td(ts_xts(uspop) ~ 1, "mean", to = "year", method = "fast"))

# quarter to daily (no indicator)
m.d.noind <- td(gdp.q ~ 1, to = "daily", method = "fast")
predict(m.d.noind)

# quarter to daily (one indicator)
m.d.stocks <- td(gdp.q ~ spi.d, method = "chow-lin-fixed", fixed.rho = 0.9)
predict(m.d.stocks)
} # }