A Randomized Sequential Procedure to Determine the Number of Factors

Abstract

This paper proposes a procedure to estimate the number of common factors k in a static approximate factor model. The building block of the analysis is the fact that the first k eigenvalues of the covariance matrix of the data diverge, whilst the others stay bounded. On the grounds of this, we propose a test for the null that the i -th eigenvalue diverges, using a randomised test statistic based directly on the estimated eigenvalue. The test only requires minimal assumptions on the data, and no assumptions are required on factors, loadings or idiosyncratic errors. The randomised tests are then employed in a sequential procedure to determine k.

Sequential testing for structural stability in approximate factor models

“…5 years. Analysis of the whole dataset using rolling samples of size m suggests between one and two factors throughout -this result consistently follows using different procedures -namely, Trapani (2017) testing procedure and the criteria by Bai and Ng (2002) and Alessi et al (2010). Therefore, we run our sequential testing pro- 515 520 525 548 2 505 522 530 537 574 506 523 531 538 576 3 513 532 541 550 598 514 534 542 552 598 4 519 546 557 569 648 520 548 560 574 669 175 1 501 515 520 525 546 502 516 521 526 552 2 507 524 531 537 564 507 524 532 539 572 3 508 532 541 550 578 508 534 542 552 579 4 518 543 552 562 598 518 544 555 565 599 250 1 502 514 520 527 552 502 514 521 527 553 2 508 524 531 540 571 508 525 532 541 571 3 510 533 542 551 590 510 534 543 552 589 4 515 542 552 562 600 515 544 553 564 603 cedure monitoring the first four factors, thus accounting both for at most two new factors emerging and for a change in...…”

“…Proof of Lemma 1. The proof is similar to that of Lemma 1 in Trapani (2017), and therefore only the main arguments are reported. Recall the definition…”

“…if t < τ , there is no change in the (r + 1)-th eigenvalue of E (X k X ′ k ), and therefore the proof that λ (r+1) (t) is finite is exactly the same as in Trapani (2017). When t ≥ τ , consider…”

“…Given that the only thing we know is that the (r + 1)-th sample eigenvalue may be bounded or unbounded, we propose to use a randomised test in order to regularize the problem. Randomisation is a widely employed approach, dating back at least to Pearson (1950); various authors have employed different ways of introducing randomness into a statistic -see Corradi and Swanson (2006), Bandi and Corradi (2014), and Trapani (2017), among others. Our methodology is based on the same approach, but with a different scope.…”

“…by Bai (2003) and Fan, Liao and Mincheva (2013). The literature has also produced many results on the determination of the number of common factors r -see, inter alia, Bai and Ng (2002), Alessi, Barigozzi and Capasso (2010), Onatski (2010), Ahn and Horenstein (2013), and Trapani (2017). The factors in equation (1) have also proven to be very effective to forecast large datasets, overcoming the curse of dimensionality issue -see e.g.…”