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genip (Series)

Generalized indicator based interpolation

Syntax

[highOutput, info] = genip(lowInput, highFreq, order, aggregation, ...)

Input arguments

lowInput [ Series ]

Low-frequency input series that will be interpolated to the highFreq frequency using the Indicator... and hard conditions specified in Hard...

highFreq [ Frequency ]

Target frequency to which the lowInput series will be interpolated; highFreq must be higher than the date frequency of the lowInput.

order [ 0 | 1 | 2 ]

Autoregressive order of the transition equation for the dynamics of the interpolated series, and for the relationship between the interpolated series and the indicator (if included).

aggregation [ "mean" | "sum" | "first" | "last" | numeric ]

Type of aggregation of quarterly observations to yearly observations; the aggregation can be assigned a 1-by-N numeric vector with the weights given, respectively, for the individual high-frequency periods within the encompassing low-frequency period.

Output arguments

highOutput [ Series ]

High-frequency output series constructed by interpolating the input lowInput using the dynamics of the indicator.

info [ struct ]

Output information struct with the the following fields:

.FromFreq

Original (low) frequency of the input series

.ToFreq

Target (high) frequency to which the input series has been interpolated

.LowRange

Low frequency date range from which the input series has been interpolated

.HighRange

High frequency date range to which the input series has been interpolated

.EffectiveLowRange

Low frequency range after excluding years with full conditioning level information

.StackedSystem

Stacked-time linear system (StackedSystem) object used to run the interpolation

Options

Range=Inf [ Inf | Dater ]

Low-frequency range on which the interpolation will be calculated; Inf means from the date of the first observation to the date of the last observation in the lowInput time series.

ResolveConflicts=true [ true | false ]

Resolve potential conflicts (singularity) between the lowInput obervatations and the data supplied through the HighLevel= option.

IndicatorLevel=[] [ empty | Series ]

High-frequency indicator whose dynamics will be used to interpolate the lowInput.

IndicatorModel="Difference" [ "Difference" | "Ratio" ]

Type of model for the relationship between the interpolated series and the indicator in the transition equation: "Difference" means the indicator will be subtracted from the series, "Ratio" means the series will be divided by the indicator.

Initials=@auto [ @auto | Series ]

Initial (presample) conditions for the Kalman filter; @auto means the initial condition will be extracted from the HardLevel time series; if no observations are supplied either directly through Initials or through HardLevel, the initial condition will be estimated by maximum likelihood.

HardLevel=[ ] [ empty | Series ]

Hard conditioning information; any values in this time series within the interpolation range or the presample initial condition (see also the option Initials) will be imposed on the resulting highOutput.

TransitionIntercept=0 [ numeric | @auto ]

Intercept in the transition equation; if @auto the intercept will be estimated by GLS.

Description

The interpolated lowInput is obtained from the first element of the state vector estimated using the following quarterly state-space model estimated by a Kalman filter:

State transition equation

\[ \left(1 - L\right)^k \hat x_t = v_t \]

where \(\hat x_t\) is a transformation of the unobserved higher-frequency interpolated series, \(x_t\), depending on the option Indicator.Model, and \(v_t\) is a transition error with constant variance. The transformation \(\hat x_t\) is given by:

  • \(\hat x_t = x_t\) if no indicator is specified;

  • \(\hat x_t = x_t - q_t\) if an indicator \(q_t\) is entered through Indicator.Level= and Indicator.Model="Difference";

  • \(\hat x_t = x_t / q_t\) if an indicator \(q_t\) is entered through Indicator.Level= and Indicator.Model="Ratio";

\(L\) is the lag operator, \(k\) is the order of differencing specified by order.

Measurement equation

\[ y_t = Z x_t \]

where

  • \(y_t\) is a measurement variables containing the lower-frequency data placed in the last (fourth) quarter of every year; in other words, only every fourth observation is available, and the three in between are missing

  • \(x_t\) is a state vector consisting of \(N\) elements, where \(N\) is the number of high-frequency periods within one low-frequency period: the unobserved high-frequency lags \(t-N, \dots, t-1, t\).

  • \(Z\) is a time-invariant aggregation matrix depending on aggregation:

  • \(Z=[1, 1, 1, 1]\) for aggregation="sum",

  • \(Z=[1/4, 1/4, 1/4, 1/4]\) for aggregation="average",
  • \(Z=[0, 0, 0, 1]\) for aggregation="last",
  • \(Z=[1, 0, 0, 0]\) for aggregation="first",

  • or a user supplied 1-by-\(N\) vector

  • \(w_t\) is a vector of measurement errors associated with soft conditions.

Examples