INCLANTIAO.RPF

INCLANTIAO.RPF is a replication for an example from Inclan and Tiao(1994). Most of the work is done by the @ICSS procedure. The Inclan-Tiao algorithm searches for structural breaks in variance. Note that these are exogenous changes in variance (that is, the model itself does not explain why the variances are different, just identifies possible change points), unlike a GARCH model where the variance changes endogenously with the data. However, if a GARCH model is correct, you are likely to find "breaks" using @ICSS or if exogenous breaks are correct, you will probably find "GARCH" behavior if you look for it.

@ICSS should be applied only to series which are reasonably assumed to be not serially correlated and to have a stationary mean. In this case, the data input to it are the log returns for IBM.

@ICSS(subsamples=ss) ldiff 2 369

analyzes the data and returns the subsamples in the 2 x (identified breaks + 1) RECT[INTEGER] SS. The number of breaks is not set in advance—the algorithm keeps working until there are no further significant breaks at the input significance level (which is .05 by default).

The remainder of the program sets up a graph which shows +/- two standard deviation bands for each subsample around the (assumed) common mean for the sample. This uses SSTATS to compute the second moment of the data over a particular subsample (the mean of the squared de-meaned values). Note that you wouldn't use STATISTICS for this because that would extract a separate mean from this sample. (You could use STATISTICS with NOCENTER).

do i=1,%cols(ss)

sstats(mean) ss(1,i) ss(2,i) demean^2>>svar

set upper ss(1,i) ss(2,i) = mean+2.0*sqrt(svar)

set lower ss(1,i) ss(2,i) = mean-2.0*sqrt(svar)

end do i

The GRAPH is done with a line graph for the data, with "step" graphs for the upper and lower bounds (which change only at a few locations and are otherwise flat). OVCOUNT=2 and OVSAME means that the last two series are covered by the STEP style and the original data and the upper and lower bounds are done using the same scale.

graph(overlay=step,ovcount=2,ovsame) 3

# ldiff

# upper / 2

# lower / 2

Note that the entry numbers are off by one from those given in the article. This is because RATS is basing entry numbers on the start of the original stock price series, not on the start of the log differenced data.

Full Program

data(unit=input) 1 369 ibm

460 457 452 459 462 459 463 479 493 490 492 498 499 497 496

490 489 478 487 491 487 482 479 478 479 477 479 475 479 476

476 478 479 477 476 475 475 473 474 474 474 465 466 467 471

471 467 473 481 488 490 489 489 485 491 492 494 499 498 500

497 494 495 500 504 513 511 514 510 509 515 519 523 519 523

531 547 551 547 541 545 549 545 549 547 543 540 539 532 517

527 540 542 538 541 541 547 553 559 557 557 560 571 571 569

575 580 584 585 590 599 603 599 596 585 587 585 581 583 592

592 596 596 595 598 598 595 595 592 588 582 576 578 589 585

580 579 584 581 581 577 577 578 580 586 583 581 576 571 575

575 573 577 582 584 579 572 577 571 560 549 556 557 563 564

567 561 559 553 553 553 547 550 544 541 532 525 542 555 558

551 551 552 553 557 557 548 547 545 545 539 539 535 537 535

536 537 543 548 546 547 548 549 553 553 552 551 550 553 554

551 551 545 547 547 537 539 538 533 525 513 510 521 521 521

523 516 511 518 517 520 519 519 519 518 513 499 485 454 462

473 482 486 475 459 451 453 446 455 452 457 449 450 435 415

398 399 361 383 393 385 360 364 365 370 374 359 335 323 306

333 330 336 328 316 320 332 320 333 344 339 350 351 350 345

350 359 375 379 376 382 370 365 367 372 373 363 371 369 376

387 387 376 385 385 380 373 382 377 376 379 386 387 386 389

394 393 409 411 409 408 393 391 388 396 387 383 388 382 384

382 383 383 388 395 392 386 383 377 364 369 355 350 353 340

350 349 358 360 360 366 359 356 355 367 357 361 355 348 343

330 340 339 331 345 352 346 352 357

set ldiff = log(ibm/ibm{1})

@ICSS(subsamples=ss) ldiff 2 369

* Pull the common mean out of the data.

diff(center) ldiff / demean

compute mean=%mean

* Compute the variance (sum of squares/T) for each of the subsamples,

* and create upper and lower 2 standard deviation bounds (relative to

* the common mean).

do i=1,%cols(ss)

sstats(mean) ss(1,i) ss(2,i) demean^2>>svar

set upper ss(1,i) ss(2,i) = mean+2.0*sqrt(svar)

set lower ss(1,i) ss(2,i) = mean-2.0*sqrt(svar)

end do i

graph(overlay=step,ovcount=2,ovsame,footer="ICSS Analysis of Variance Breaks") 3

# ldiff

# upper / 2

# lower / 2

Output

ICSS Analysis of Series LDIFF

Using significance level 0.05

Breakpoints found at entries

236

280

Graph