B) OBLIGACIÓN TRIBUTARIA
B.3 EL SUJETO PASIVO
II.4. EL NOTARIO COMO SUJETO PASIVO DEL IMPUESTO AL VALOR AGREGADO (IVA)
It is commonly understood that at site s, one experiences a heatwave event if the temperatures are unusually high (relative to the average seasonal temperature) for a prolonged period of time. Depending on the temperature amplitude and tempo- ral duration of the event one encounters different types of heatwaves. We consider heatwave events identified by ∆ consecutive days of daily maximum temperatures exceeding a level u0. To be able to account for trend in temperature patterns (be it
seasonal or global), we need to consider a threshold u0 that varies slowly with the
season. Alternatively, one can consider deviations from the mean temperature curve of a station with respect to its standard devation curve.
Specifically, let Xt be the daily temperature maxima over a period of ∆ = 7
consecutive days starting on day k. Several sophisticated techniques, both parametric Ramsay et al. (2005) and non parametric Ferraty and Vieu (2006) techniques exist for modeling of functional curves, we rather use the one motivated from Section 3.5 of Ramsay et al. (2005). We assume the following representation for the mean of the
time series Xt(s): E(Xt) := µ(t) = γ(µ)+ m(µ)1 X i=1 α(µ)i φ(µ)i (t) | {z } A(µ)(t) + m(µ)2 X j=1 βj(µ)ωj(µ)(t) | {z } B(µ)(t) (4.1)
where γ(µ) is a constant term and α(µ) i and β
(µ)
j are the coefficients corresponding to
the covariates of seasonal fluctuations and global trends respectively.
The functions φ(µ)i , i = 1, · · · , m(µ)1 denote a 365- periodic cubic7 spline basis
representation8 of the range 1, 2, · · · , 365T with T = 100 with m = m(µ)
1 degrees of
freedom. Figure 4.1 left panel displays a plot of 365-periodic cubic splines with 3 degrees of freedom when evaluated on the time span 1986-1990. The quantity A(µ)(t)
thus captures the portion of the mean curve that may be attributed to seasonal variations.
On the other hand, the functions ωi(µ), i = 1, · · · , m(µ)2 denote a cubic spline basis representation9 of the range 1, · · · , 365T . Figure 4.1 middle panel displays a plot of
cubic splines with 3 degrees of freedom when evaluated on the time span 1911-2010. B(µ)(t) thus captures the portion of the mean curve that may be attributed to slowly
varying trends over time like global warming, shifting weather conditions etc.
The residuals: ˜Xt= Xt− ˆµ(t) obtained from the linear regression model in (4.1)
provides a way for the estimation of the standard error curve. We propose the fol-
7Knott (2000)
8Since seasonal patterns are expected to repeat themselves after a span of 365 days, 365-periodic
spline functions with m degrees of freedom are constructed as in Bojanov et al. (1993): φi(a) = φ(a + 365), a = 1, · · · , 365, i = 1, · · · , m.
The pbs function under R package pbs (2013) automates this process periodic spline generation.
9For m degrees of freedom, the spline basis functions ω
j, j = 1, · · · , m are constructed as in
Bojanov et al. (1993) to describe the yearly trends and phenomenon like global warming, shifting weather conditions etc. The bs function under R package splines (2000) automates this process the spline generation.
lowing representation of variance of time series ˜Xt. E(log | eXt|) :≈ log(cη(t)) = γ(η)+ m(η)1 X i=1 α(η)i φ(η)i (t) | {z } A(η)(t) + m(η)2 X j=1 βjω (η) j (t) | {z } B(η)(t) (4.2)
where γ(η)is a constant term and α(η) i and β
(η)
j are the coefficients corresponding to the
covariates of seasonal fluctuations and global trends respectively. A normal approxi- mation to the time series ˜Xt provides a choice for the parameter c = 0.5298. Similar
to the model in (4.1), the functions φ(η)(t) are 365-periodic cubic splines and ω(η)(t)
are ordinary cubic splines for the basis representation of the range 1, 2, · · · , 365T with m = m(η)1 and m = m(η)2 degrees of freedom respectively. Thus, the quantities A(η)(t)
and B(η)(t) denote the portion of the standard deviation explained by seasonality and
global trend respectively.
For known values of m(µ)1 , m(µ)2 , m(η)1 and m(η)2 , the unknown coefficients in models (4.1) and (4.2) may be easily estimated using a simple linear regression approach.
Figure 4.1: Spline basis functions evaluated. Left panel corresponds to the periodic splines for seasonal activity, middle panel corresponds to splines for global activity and right panel corresponds to splines for ENSO activity
Exploratory analysis revealed little sensitivity to the choice of the parameters m(µ)1 , m(µ)2 , m(η)1 and m(η)2 . We however added an additional criterion to guard against large
choices of these quantities to en certain that meaningful information does not get lost from the time series in the process of following the mean or standard deviation too closely. We next explain the simultaneous choice of m(µ)1 , m(µ)2 .
(m∗1, m∗2) = argmin
{m1≤κ1,m2≤κ2}
n log(RSS/n) + 2(m1+ m2) (4.3)
where RSS is the residual sum of squares obtained from the linear regression fit in (4.1) with m(µ)1 = m1 and m
(µ)
2 = m2. A choice of thresholds as low as κ1 = 6 and
κ2 = 6 produced standardized z scores which were fairly stationary (see right panel
in Figure 4.3) and seemed to provide a promising choice for the remaining analysis. The quantities, m(η)1 and m(η)2 are also estimated according to the criterion in (4.3) but for the linear model in (4.2). Section 4.6 further sheds some light on the choice of the parameters κ1 and κ2.
Figure 4.2: Seasonal and trend components for historical temperature mean (top panel) and standard error (bottom panel) curves over the period 1911- 2010 for Ann Arbor, MI.
Figure 4.2 top panel illustrates the seasonal and trend components, A(µ)(t) and
B(µ)(t) of the mean curve µ(t) respectively. The bottom panel in 4.2 illustrates
seasonal and trend components, A(η)(t) and B(η)(t) of the standard error curve η(t).
These reported graphs are obtained from the historical daily temperature records over period 1911 − 2010 at the monitoring station of Ann Arbor in continental US. The degrees of freedom m(µ)1 , m(µ)2 , m(η)1 , m(η)2 are chosen according to the criterion in (4.3) with κ1 = 6 and κ2 = 6.