2:41 a.m.
PS: JMULTI computes eigenvalues of the reverse characteristic polynomial
which must be greater than one for stability, while Gretl computes inverse
roots which must be less than 1.
On Tue, Dec 13, 2011 at 1:33 PM, Muheed Jamaldeen <mj.myworld(a)gmail.com>wrote:
Hi Peter,
Yes. USGDP is part of the 11 variables. I didn't want to clutter the list
with the rest of the tests and other information.
I am estimating the reduced form (near) VAR and SVAR on JMULTI whilst
using Gretl for preliminary testing. JMULTI computes modulus of the
eigenvalues of the reverse characteristic polynomial which are all greater
than one (which is the stability condition, implying stationarity).
Let me explain what I mean by stability implying stationarity using
Lutekepohl's text:
" The process is stable if det(IK − A1z − ·· ·− Apz p) = 0 for |z| ≤ 1,
(3.2) that is, the polynomial defined by the determinant of the
autoregressive operator has no roots in and on the complex unit circle. On
the assumption that the process has been initiated in the infinite past (t
= 0,±1,±2, . . .), it generates stationary time series that have
time-invariant means, variances, and covariance structure."
He makes a similar argument here:
http://books.google.com.au/books?id=muorJ6FHIiEC&q=Stable#v=snippet&a...
See the proposition in page 25.
That makes sense to me because regardless of what the univariate Data
Generating Process (DGP) suggests, the description of the process by each
VAR equation is stable, implying stationarity. If this is WITHOUT a trend
term, then is there a case for a trend term? That is the crux of the
question. The distinction is that the DGP may not be stationary, but the
VAR process is. So there is no problem if the VAR process is stable (and
thereby stationary) without a trend term.
I agree with your small sample point. But my hands are tied there! lol.
Also, for consistency I computed the VAR inverse roots on Gretl. They are
all inside the unit circle. Implying stability of the VAR process.
roots (real, imaginary, modulus, frequency)
1: (-0.6179, 0.4963, 0.7925, 0.3923)
2: (-0.6179, -0.4963, 0.7925, -0.3923)
3: (-0.7550, 0.0000, 0.7550, 0.5000)
4: (-0.7032, 0.1521, 0.7195, 0.4661)
5: (-0.7032, -0.1521, 0.7195, -0.4661)
6: (-0.4947, 0.6105, 0.7857, 0.3584)
7: (-0.4947, -0.6105, 0.7857, -0.3584)
8: (-0.5062, 0.3974, 0.6435, 0.3941)
9: (-0.5062, -0.3974, 0.6435, -0.3941)
10: (-0.5245, 0.2641, 0.5872, 0.4258)
11: (-0.5245, -0.2641, 0.5872, -0.4258)
12: (-0.2243, 0.7003, 0.7354, 0.2993)
13: (-0.2243, -0.7003, 0.7354, -0.2993)
14: (-0.3287, 0.5380, 0.6305, 0.3373)
15: (-0.3287, -0.5380, 0.6305, -0.3373)
16: ( 0.3527, 0.8513, 0.9215, 0.1875)
17: ( 0.3527, -0.8513, 0.9215, -0.1875)
18: (-0.0961, 0.5961, 0.6038, 0.2754)
19: (-0.0961, -0.5961, 0.6038, -0.2754)
20: ( 0.2624, 0.7219, 0.7681, 0.1945)
21: ( 0.2624, -0.7219, 0.7681, -0.1945)
22: ( 0.2109, 0.6545, 0.6876, 0.2004)
23: ( 0.2109, -0.6545, 0.6876, -0.2004)
24: (-0.2329, 0.0000, 0.2329, 0.5000)
25: (-0.1000, 0.0000, 0.1000, 0.5000)
26: ( 0.6966, 0.6594, 0.9592, 0.1206)
27: ( 0.6966, -0.6594, 0.9592, -0.1206)
28: ( 0.4453, 0.5594, 0.7149, 0.1430)
29: ( 0.4453, -0.5594, 0.7149, -0.1430)
30: ( 0.7258, 0.4395, 0.8485, 0.0866)
31: ( 0.7258, -0.4395, 0.8485, -0.0866)
32: ( 0.8118, 0.3423, 0.8810, 0.0635)
33: ( 0.8118, -0.3423, 0.8810, -0.0635)
34: ( 0.9206, 0.2748, 0.9608, 0.0462)
35: ( 0.9206, -0.2748, 0.9608, -0.0462)
36: ( 0.9928, 0.0788, 0.9959, 0.0126)
37: ( 0.9928, -0.0788, 0.9959, -0.0126)
38: ( 0.9958, 0.0000, 0.9958, 0.0000)
39: ( 0.8962, 0.1124, 0.9032, 0.0198)
40: ( 0.8962, -0.1124, 0.9032, -0.0198)
41: ( 0.5908, 0.2922, 0.6591, 0.0731)
42: ( 0.5908, -0.2922, 0.6591, -0.0731)
43: ( 0.5367, 0.2025, 0.5736, 0.0574)
44: ( 0.5367, -0.2025, 0.5736, -0.0574)
Hope that helps.
Cheers,
Mj
On Tue, Dec 13, 2011 at 12:51 PM, Summers, Peter <psummers(a)highpoint.edu>wrote:
> I'm confused too.
>
> MJ, is the (log) level of GDP one of the 11 series in your VAR? If so,
> then based on the unit root tests you showed earlier, it is not "stable
> (and stationary) without a trend." On the contrary, it has a unit root --
> and is therefore non-stationary -- whether or not a deterministic trend is
> included in the dgp.
>
> In other words, including the level of GDP in your reduced-form VAR
> renders it non-stationary.
>
> Back to the potential small-sample issue: a VAR with 11 variables and 4
> lags has 44 parameters per equation, not counting a constant (or trend?!).
> There are also 55 parameters in the covariance matrix. With 100
> observations per series, you're asking quite a lot of your data set. Even
> if you knew the covariance matrix for sure, you'd have just over 2
> obs/parameter for estimating the dynamics. I don't think that's asymptotic
> yet, but I could be wrong ;-)
>
>
> ________________________________
> From: gretl-users-bounces(a)lists.wfu.edu [
> gretl-users-bounces(a)lists.wfu.edu] on behalf of Dr RJF Hudson [
> rjfhud(a)powerup.com.au]
> Sent: Monday, December 12, 2011 8:28 PM
> To: Gretl list
> Subject: Re: [Gretl-users] Deterministic trend in VAR
>
> Greetings all
> Have to say I'm getting confused, here.
> I'd be appreciative please if somebody would tell me please
> what this means "the reduced form".... of what?
> Also if a set is stable as you say, and to produce its stationarity you
> are confident that you haven't
> squelched out important information from the data by differencing etc,
> what's the reason to introduce trend information and then trust inferences
> from the results ?
> Trend in their Unit Roots?
> I'm cool
> rest easy
> Richard Hudson
>
> Dr RJF Hudson Qld Australia
> rjfhud@powerup.com.au<mailto:rjfhud@powerup.com.au>
> ----- Original Message -----
> From: Muheed Jamaldeen<mailto:mj.myworld@gmail.com>
> To: Gretl list<mailto:gretl-users@lists.wfu.edu>
> Sent: Tuesday, December 13, 2011 10:59 AM
> Subject: Re: [Gretl-users] Deterministic trend in VAR
>
> You're right about the VAR not being stable if USGDP were the only series
> in the model. Well, the VAR is a 11 variable VAR (4). The 11 variables are
> GDP and macroeconomic variables.
>
> I am testing the impact of cash rate innovations on GDP. The question is,
> if the reduced form is stable (and stationary) WITHOUT a trend, should one
> include a trend when the univariate tests suggest that SOME of the series
> may have trend in their unit roots.
>
> Hope that makes sense?
>
>
> On Tue, Dec 13, 2011 at 11:46 AM, Summers, Peter <psummers(a)highpoint.edu
> <mailto:psummers@highpoint.edu>> wrote:
> MJ,
>
> You're right that the unit root tests are telling you that you have a
> unit root in at least one series.
>
> I'm confused about what your VAR looks like though (and maybe the rest of
> the list is too). If this is one of the series in your VAR, then it's not
> stable/stationary, by definition. That is, the lag operator polynomial will
> have at least one root on the unit circle. My earlier answer assumed that
> your unit root & cointegration tests ruled out both, but now it seems
> that's not the case.
>
> Relating to ths, how many series do you have in your VAR? My feeling is
> that 100 obs per series isn't really a lot, especially if you're trying to
> sort out issues related to deterministic vs stochastic trends,
> cointegration vs none, etc.
>
> At this point I'd suggest a) reading the gretl manual and/or your
> favorite reference on VARs & VECMs, and/or b) providing some more detail
> about what you're trying to do.
>
> PS
> ________________________________
> From: gretl-users-bounces(a)lists.wfu.edu<mailto:
> gretl-users-bounces(a)lists.wfu.edu> [gretl-users-bounces(a)lists.wfu.edu
> <mailto:gretl-users-bounces@lists.wfu.edu>] on behalf of Muheed
> Jamaldeen [mj.myworld@gmail.com<mailto:mj.myworld@gmail.com>]
> Sent: Monday, December 12, 2011 6:59 PM
> To: Gretl list
> Subject: Re: [Gretl-users] Deterministic trend in VAR
>
> Peter,
>
> I have 100 observations in the model. So small samples may or may not be
> an issue. I am wondering if the deterministic trend is an issue at all
> because the VAR is stable implying stationarity of the described process in
> each equation WITHOUT the trend (i.e. the polynomial defined by the
> determinant of the autoregressive operator has no roots in and on the
> complex unit circle without the time trend term).
>
> The ADF tests suggest that we cannot reject the trend term. Let me show
> you an example. Following is the ADF tests for logged US GDP.
>
> Monte Carlo studies suggest that choosing the lag order (p) of the unit
> root tests according to the formula: Int {12(T /100)1/ 4} so the lag order
> is 12 with 100 observations.
>
> test without constant
> test statistic: tau_nc(1) = 2.13551
> asymptotic p-value 0.9927
>
> test with constant
> test statistic: tau_c(1) = -1.28148
> asymptotic p-value 0.6405
>
> with constant and trend
> test statistic: tau_ct(1) = -0.728436
> asymptotic p-value 0.9702
>
> Following is the estimate for the trend term in the last ADF regression.
>
> coefficient std. error t-ratio p-value
> -------------------------------------------------------------
> time 0.000200838 0.000317669 0.6322 0.5292
>
> So all three tests are saying that I cannot reject the null of unit root.
> Including I(1) variables in an unrestricted VAR is fine as Lutekepohl and
> Toda and Yammoto have demonstrated. It's a question of whether a trend term
> is to be included. I am inclined to think not because the VAR is stable
> WITHOUT a trend.
>
> Thoughts?
>
> Cheers,
>
> Mj
>
> On Tue, Dec 13, 2011 at 1:17 AM, Summers, Peter <psummers(a)highpoint.edu
> <mailto:psummers@highpoint.edu><mailto:psummers@highpoint.edu<mailto:
> psummers(a)highpoint.edu>>> wrote:
> MJ,
>
> If your data have deterministic trends, then unit root tests should pick
> that up (though there may be a problem in small samples). If you include a
> trend but the dgp is stationary, then a t-test should conclude that the
> trend coefficient is zero. Presumably your unit root tests reject the null,
> right?
>
> From: gretl-users-bounces(a)lists.wfu.edu<mailto:
> gretl-users-bounces(a)lists.wfu.edu><mailto:
> gretl-users-bounces(a)lists.wfu.edu<mailto:
> gretl-users-bounces(a)lists.wfu.edu>> [mailto:
> gretl-users-bounces(a)lists.wfu.edu<mailto:
> gretl-users-bounces(a)lists.wfu.edu><mailto:
> gretl-users-bounces(a)lists.wfu.edu<mailto:
> gretl-users-bounces(a)lists.wfu.edu>>] On Behalf Of Muheed Jamaldeen
> Sent: Monday, December 12, 2011 5:52 AM
> To: Gretl list
> Subject: [Gretl-users] Deterministic trend in VAR
>
> Hi all,
> Just a general VAR related question. When is it appropriate to include a
> deterministic time trend in the reduced form VAR? Visually some of the data
> series (not all) look like they have trending properties. In any case, does
> the inclusion of the time trend matter if the process is stable and
> therefore stationary (i.e. the polynomial defined by the determinant of the
> autoregressive operator has no roots in and on the complex unit circle)
> without the time trend term. Other than unit root tests, is there a better
> way to test whether the underlying data generating process has a stochastic
> or deterministic process?
>
> I am mainly interested in the impulse responses.
>
> Cheers,
>
> Mj
>
>
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