<p>Consider a dataset of N time series and T observation periods.
Assume each series <span class="math-container">$x_t$</span> is generated from a <strong>single</strong> (unobserved) <strong>common factor</strong> <span class="math-container">$f_t$</span> following this model:</p>
<p><span class="math-container">$$
X_t = \Lambda f_t + \epsilon_t, \quad \text{with} \quad \epsilon_t \sim N(0, \sigma^2_\epsilon) \\
f_t = \rho f_{t-1} + v_t, \quad \text{with} \quad v_t \sim N(0, \sigma^2_v)  \quad and \quad |\rho|&lt;1
$$</span></p>
<p>where <span class="math-container">$X_t$</span> is a <span class="math-container">$N \times 1$</span> column vector of <strong>observations</strong>, <span class="math-container">$\Lambda$</span> is a <span class="math-container">$N \times 1$</span> vector of <strong>&quot;factor loadings&quot;</strong> indicating how the common factor influences each observed series and <span class="math-container">$\epsilon_t$</span> is <span class="math-container">$N \times 1$</span> vector of <strong>gaussian innovation terms</strong> with no serial or cross-sectional correlation.</p>
<p>The common factor <span class="math-container">$f_t$</span> follows an <strong>AR(1) dynamic</strong> and in both equations the idiosyncratic terms, <span class="math-container">$\epsilon_t$</span> and <span class="math-container">$v_t$</span>, have finite variance.</p>
<p>This should be a very simple dynamic factor model setup.</p>
<p><strong>Under which conditions the first Principal Component of the observation matrix should estimate consistently the common factor?</strong></p>
<p>I am interested in testing the performance of different factor extraction methods within small samples and I have built a Monte Carlo simulation for this purpose. However, I am observing that as the cross-section (N) of my (artificial) dataset increases, the estimates of the common factor deteriorates in terms of higher <strong>Mean Squared Error</strong> (MSE) between the estimated factor series and the true one. I'd like to know why.</p>
<p>On the contrary, I do observe that as T increases the estimates become more precise, with a lower MSE.</p>
<p>To obtain the artificial dataset I first generate an AR(1) series with gaussian error terms to play the role of the common factor <span class="math-container">$f_t$</span>. Then, I draw randomly N factor loadings to compute N linear transformations of the original latent series contaminated with other gaussian error terms. Such that, e.g., the first observed series is obtained as:</p>
<p><span class="math-container">$$
x_{1,t} = \lambda_1 f_t + \epsilon_{1,t}
$$</span></p>
<p>Thank you very much in advance for your replies!</p>

CC BY-SA 4.0
2024-11-18T08:38:29.203Z
I think the standard conditions are based on large T asymptotics. So from that point of view, it is no surprise that increasing N and therefore the complexity of the correlational object of interest, becomes harder to estimate. However, maybe you should expand on what you mean by 'deteriorating estimates'.
Jesper Hybel
CC BY-SA 4.0
2024-11-18T10:18:16.953Z
Thank you @JesperHybel, I edited the question by specifying that the precision of factor estimates is computed as the Mean Squared Error between the estimated series and the true latent factor series.

Regarding the consistency conditions, in [this](https://www.princeton.edu/~mwatson/papers/Stock_Watson_JASA_2002.pdf) article by Stock and Watson it is stated that both T and N must tend to infinity.
For this reason I don't understand why as N increases my common factor estimates get worse.
NicGeraci
CC BY-SA 4.0
2024-11-20T16:41:11.423Z
Are you sure you are implementing it correctly? For example, how do you make sure that $\Lambda^\top\Lambda/N \rightarrow I_r$ as N increase?
Jesper Hybel
<econometrics><time-series><nowcasting><kalman>
Under which conditions does PCA consistently estimate latent factors in a Dynamic Factor Model?
NicGeraci
# Under which conditions does PCA consistently estimate latent factors in a Dynamic Factor Model?

(Asked by: username_0 on 2024-11-18)

Consider a dataset of N time series and T observation periods.
Assume each series $x\_t$ is generated from a **single** (unobserved) **common factor** $f\_t$ following this model:

$$
X\_t = \Lambda f\_t + \epsilon\_t, \quad \text{with} \quad \epsilon\_t \sim N(0, \sigma^2\_\epsilon) \\
f\_t = \rho f\_{t-1} + v\_t, \quad \text{with} \quad v\_t \sim N(0, \sigma^2\_v) \quad and \quad |\rho|<1
$$

where $X\_t$ is a $N \times 1$ column vector of **observations**, $\Lambda$ is a $N \times 1$ vector of **"factor loadings"** indicating how the common factor influences each observed series and $\epsilon\_t$ is $N \times 1$ vector of **gaussian innovation terms** with no serial or cross-sectional correlation.

The common factor $f\_t$ follows an **AR(1) dynamic** and in both equations the idiosyncratic terms, $\epsilon\_t$ and $v\_t$, have finite variance.

This should be a very simple dynamic factor model setup.

**Under which conditions the first Principal Component of the observation matrix should estimate consistently the common factor?**

I am interested in testing the performance of different factor extraction methods within small samples and I have built a Monte Carlo simulation for this purpose. However, I am observing that as the cross-section (N) of my (artificial) dataset increases, the estimates of the common factor deteriorates in terms of higher **Mean Squared Error** (MSE) between the estimated factor series and the true one. I'd like to know why.

On the contrary, I do observe that as T increases the estimates become more precise, with a lower MSE.

To obtain the artificial dataset I first generate an AR(1) series with gaussian error terms to play the role of the common factor $f\_t$. Then, I draw randomly N factor loadings to compute N linear transformations of the original latent series contaminated with other gaussian error terms. Such that, e.g., the first observed series is obtained as:

$$
x\_{1,t} = \lambda\_1 f\_t + \epsilon\_{1,t}
$$

Thank you very much in advance for your replies!

**Tags:** econometrics, time-series, nowcasting, kalman
> **Comments:**
> **username_1**: I think the standard conditions are based on large T asymptotics. So from that point of view, it is no surprise that increasing N and therefore the complexity of the correlational object of interest, becomes harder to estimate. However, maybe you should expand on what you mean by 'deteriorating estimates'.  
> **username_0**: Thank you @JesperHybel, I edited the question by specifying that the precision of factor estimates is computed as the Mean Squared Error between the estimated series and the true latent factor series.  Regarding the consistency conditions, in [this](https://www.princeton.edu/~mwatson/papers/Stock_Watson_JASA_2002.pdf) article by Stock and Watson it is stated that both T and N must tend to infinity. For this reason I don't understand why as N increases my common factor estimates get worse.  
> **username_1**: Are you sure you are implementing it correctly? For example, how do you make sure that $\Lambda^\top\Lambda/N \rightarrow I_r$ as N increase?  

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## 0 answers

